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Upgrade of Algebraic Structure of Ring

Upgrade of Algebraic Structure of Ring Fuzzy Inf. Eng. (2009)2:219-228 DOI 10.1007/s12543-009-0017-x ORIGINAL ARTICLE Bing-xue Yao · Yu-bin Zhong Received: 16 January 2009/ Revised: 10 May 2009/ Accepted: 20 May 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In order to construct nonregular power rings, the concept of regular semi- ideal with respect to a subring is introduced. Then several constructive theorems of HX ring and power ring are established and some examples of non-regular power rings are constructed which contain some nonregular HX rings. Keywords HX rings · Power ring · Regular power ring · Nontrivial HX ring · Nontrivial power ring 1. Introduction With the successful upgrade of algebraic structure of group, many researchers con- sidered the upgrade of algebraic structure of some other algebraic systems, in which the ring was considered first. In 1988, Professor Li proposed the concept of HX ring [1] and derived some of their properties. Then Professor Zhong [2] gave the struc- tures of HX-ring on a class of ring and cited an example of nontrivial HX ring [3], proving the existence of nontrivial HX ring. In this paper, we establish a series of structure theorems and constructive theorems of power ring and give some examples of nontrivial HX ring and nontrivial power ring. Unless otherwise statement, (R,+,·) will always stand for an associative ring with zero element 0. 2. Concepts of HX Ring and Power Ring Let P (R) = P(R)\{Φ}, A, B ∈ P (R),we define the following operations [1]: 0 0 A+ B = {a+ b|a ∈ A, b ∈ B}, (2.1) AB = {ab|a ∈ A, b ∈ B}. (2.2) Bing-xue Yao School of Mathematics Sciences, Liaocheng University, Liaocheng 252059, P.R.China e-mail: yaobingxue@lcu.edu.cn Yu-bin Zhong () School of Mathematics and Information Sciences,Guangzhou University, Guangzhou 510006, P.R.China e-mail: Zhong yb@163.com 220 Bing-xue Yao · Yu-bin Zhong (2009) For convenience, we write a + B and aB and Ab instead of {a} + B and {a}B and A{b}, respectively. Clearly, let A, B, C ∈ P (R), then (I) A+ B = B+ A, (II) (A+ B)+ C = A+ (B+ C), (III) (AB)C = A(BC), (IV) (A(B+ C) ⊆ AB+ AC, (B+ C)A ⊆ BA+ BC, (V) A ⊆ B ⇒ A+ C ⊆ B+ C, AC ⊆ BC, CA ⊆ CB. Definition 2.1 [1] LetR be a nonempty subset of P (R) such thatR forms a ring for the operations (2.1) and (2.2). Then R is called an HX ring on R whose zero element is denoted by Q and the negative element of A∈R is denoted by−A. Example 2.2 Let S be a subring of R. Then{{S}|s ∈ S} is an HX ring on R.If A is a nonempty subset of R such that A+ A = AA = A, then {A} is also an HX ring. These HX rings are trivial HX rings. Example 2.3 [2] Let C be the set of all nonzero complex numbers. The operations ”⊕” and ”⊗”of C are defined as follows: ln|b| 0 a⊕ b = ab, a⊗ b = |a| ,∀a, b ∈ C . Then (C ,⊕,⊗) forms a ring. Let I = (1,+∞) and H = {1,−1, i,−i}. ThenR = {a⊕ I|a ∈ H is an HX ring on(C ,⊕,⊗). Remark A quotient ring of a given ring R,may not be an HX ring, because the multiplication of the quotient ring is not coincident with the operation (2.2), although its addition coincides with the operation (2.1). Example 2.4 [4] Let Z denote the ordinary integral number ring and letI =< 10 > be the ideal generated by 10. Then the quotient ring (Z/I,+,◦)of Z with respect to I satisfies: (z + I)+ (z + I) = (z + z )+ I, (z + I)◦ (z + I) = z z + I,∀z , z ∈ Z. 1 2 1 2 1 2 1 2 1 2 Let z = 2, z = 4. Then 1 2 (z + I)(z + I) = {8+ 20m+ 40n+ 100mn|m, n ∈ Z}, 1 2 z z + I = {8+ 10m|m ∈ Z}. 1 2 Clearly, (z + I)◦ (z + I) = z z + I  (z + I)(z + I), 1 2 1 2 1 2 since 18 ∈ z z + I and 18  (z + I)(z + I). This means that (Z/I,+,◦) is not an HX 1 2 1 2 ring. In fact, if I is an ideal of R, then for all r , r ∈ R we have 1 2 (r + I)(r + I) ⊆ r r + I. 1 2 1 2 However, in general, “⊆” can not be replaced by “=”. Definition 2.5 [5] Let R be a nonempty subset of P (R) with two operations, ad- dition “+” and multiplication “◦”, such that A + B = {a + b|a ∈ A, b ∈ B} and Fuzzy Inf. Eng. (2009) 2: 219-228 221 A◦ B ⊇ AB for all A, B∈R.If (R,+,◦) forms a ring, then (R,+,◦) is called a power ring on R whose zero element is denoted by Q. Clearly, (I) if (R,+,◦) is a power ring on R, then (R,+) is a power group on (R,+); (II) all quotient rings of every subring of R and all HX rings on R are also power rings on R. Example 2.6 Let R be the ordinary real number ring and let E = (0,+∞),R = {a+ E|a ∈ R}. We define operations in R as follows: (a+ E)+ (b+ E) = (a+ b)+ E, (a+ E)⊗ (b+ E) = ab+ E,∀a, b ∈ R. Then (R,+,⊗) is a ring but not a power ring because of (−1+ E)⊗ (2+ E) = −2+ E and −2+ E  (−1+ E)(2+ E). Definition 2.7 All quotient rings of every subring of R and all trivial HX rings on R are called trivial power rings on R. 3. Structure of Power Ring Let (R,+,◦) be a power ring on R. Then R = ∪{A|A ∈R} is called a basic element set of (R,+,◦). Definition 3.1 Let S be a nonempty subset of R such that S + S ⊆ S and S S ⊆ S .Then S is called a subsemiring of R. Theorem 3.2 Let (R,+,◦) be a power ring on R. Then (I) A ⊆ B⇒−B⊆−A,∀A, B∈R, (II) AQ∪ QA ⊆ Q,∀A∈R, (III) R is a subsemiring of R, (IV) Q is a subsemiring of R. Theorem 3.3 Let (R,+,◦) be a power ring on R. Then 0∈R ⇒ 0 ∈ Q. Theorem 3.4 Let (R,+,◦) be a power ring on R. Then (I) A ⊆ B ⇒ AB = BA = AA = BB,∀A, B∈R, (II) Q ⊆ A ⇒ AB = BA = Q,∀B∈R. Definition 3.5 Let (R,+,◦) be a power ring on R and let A ∈R. Then A = {a ∈ A|(−a) ∈ (−A)} is called the kernel of A. Obviously, the concept of kernel coincides with that of power group [6,7]. Definition 3.6 Let (R,+,◦) be a power ring on R. If 0 ∈ Q. Then (R,+,◦) is called a regular power ring; if Q is a subring of R, then (R,+,◦) is called a uniform power ring. In particular, if R is an HX ring on R, we have the concepts of regular HX ring and uniform HX ring. If (R,+,◦) is a regular (uniform) power ring, then we say that (R,+,◦) is regular (uniform). Clearly, if (R,+,◦) is a regular (uniform) power ring on R, then (R,+) is a regular (uniform) power group on (R,+), so we have the following conclusions similar to those of a power group. Theorem 3.7 Let (R,+,◦) be a regular power ring on R. Then A  Φ for all A∈R. 222 Bing-xue Yao · Yu-bin Zhong (2009) Theorem 3.8 Let (R,+,◦) be a power ring on R. If there exists A ∈R such that A  Φ, then (R,+,◦) is a regular power ring. Theorem 3.9 Let (R,+,◦) be a regular power ring on R. Then for all A, B∈R,we have (I) A+ B = A+ B, (II) AB∪ AB ⊆ A◦ B. Theorem 3.10 Let (R,+,◦) be a regular power group on R. Then for all A, B∈R, we have (I) QA∪ AQ ⊆ Q, (II) A B ⊆ A◦ B. In general, “A B = A◦ B ” does not hold even if in a regular power ring. 3×3 Example 3.11 Let R be an ordinary ring of 3× 3 matrix over a real number field and let ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a 00⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ H = ⎪ 000⎪ |a ∈ R , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I = ⎪ ⎪ |b, c, x ∈ R, b  0, c  0 . ⎪ 0 bx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 00 c Then ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I + I = I, HI = IH = ⎪ 000⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ Let b , b , c , c , x , x ∈ R such that b , b , c , c  0. Then 1 2 1 2 1 2 1 2 1 2 ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 00 0 ⎪ ⎪ 00 0 ⎪ ⎪ 00 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 b x ⎪ ⎪ 0 b x ⎪ = ⎪ 0 b b b x + c x ⎪ ∈ I, ⎪ 1 1 ⎪ ⎪ 2 2 ⎪ ⎪ 1 2 1 2 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 c 00 c 00 c c 1 2 1 2 that is, II ⊆ I. Let b, c, x ∈ R such that b, c  0. Then ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ 000⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 bx⎪ = ⎪ 0 bx⎪ ⎪ 010⎪ ∈ II, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 c 00 c 001 that is, I ⊆ II. So, II = I. 3×3 Now we show that {h+ I|h ∈ H} is a nontrivial HX ring on R . Let h , h ∈ H. Then 1 2 (h + I)(h + I) ⊆ h h + h I + Ih + II = h h + I. 1 2 1 2 1 2 1 2 Conversely, if x ∈ I, then there exist x , x ∈ I, such that x = x x . Sowehave 1 2 1 2 h h + x = h h + x x = (h + x )(h + x ) ∈ (h + I)(h + I). 1 2 1 2 1 2 1 1 2 2 1 2 Fuzzy Inf. Eng. (2009) 2: 219-228 223 That is, h h + I ⊆ (h + I)(h + I). Thus we have 1 2 1 2 (h + I)(h + I) = h h + I. 1 2 1 2 3×3 Hence {h+ I|h ∈ H} is an HX ring on R . Moreover, {h+ I|h ∈ H} is a regular HX 3×3 ring because of I contains the zero element of R . It is obvious that ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ II = I = ⎪ 00 x⎪ |x ∈ R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ and ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I I = ⎪ 000⎪ , soI I  II. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ Theorem 3.12 Let (R,+,◦) be a regular power ring on R. Then R = ∪{A|A∈R} is a subring of R and Q is an ideal of R . Theorem 3.13 Let (R,+,◦) be a regular power ring on R. Then the following con- ditions are equivalent. (I) a ∈ A, (II) A = a+ Q, (III) (−A) = (−a)+ Q, (IV) A = a+ Q. Theorem 3.14 Let (R,+,◦) be a regular power ring on R. Then (I) A, B∈R⇒ A+ B = (a+ Q)+ (b+ Q) = (a+ b)+ Q, where a ∈ A, b ∈ B, (II) A, B∈R⇒ A◦ B = (a+ Q)◦ (b+ Q) = ab+ Q, where a ∈ A, b ∈ B, (III) R = {a+ Q|a ∈ R }, (IV) R /Q  (R,+,◦). Proof We only prove (II) and (IV). (II) From a ∈ A and b ∈ B we have ab ∈ A B ⊆ A◦ B, so A◦ B = ab+ Q, that is, A◦ B = (a+ Q)◦ (b+ Q) = ab+ Q. (IV) Clearly,R is a subring of R from Theorem 3.12. Let ∗ ∗ f : R →R, a → f (a) = a+ Q,∀a ∈ R . Then f is an epimorphism and ∗ ∗ ker f = {a ∈ R |a+ Q = Q} = {a ∈ R |a ∈ Q,−a ∈ Q} = Q. Hence R /Q  (R,+,◦). 224 Bing-xue Yao · Yu-bin Zhong (2009) The above theorem shows that the multiplication of a regular power ring is de- termined even though the multiplication of a power ring is not defined specifically. Later, we will show that a regular power ring is isomorphic to a quotient ring. Theorem 3.15 Let (R,+,◦) be a uniform power ring on R and let A∈R. Then (I) A = A, − − (II) −A = A , where A = {−a|a ∈ A}. Theorem 3.16 Let (R,+,◦) be a power ring on R. If there exists A ∈R, such that A = Aor−A = A , then (R,+,◦) is a uniform power ring. Theorem 3.17 Let (R,+,◦) be a uniform power ring on R. Then (I) R is a subring of R, (II) Q is an ideal of R , (III) A∈R⇒ A = a+ Q, where a ∈ A, ∗ ∗ ∗ (IV) (R,+,◦) = R /Q, where R /Q is the quotient ring of R with respect to Q. 4. Construction of HX Ring and Power Ring Upgrade of algebraic structure of ring is more difficult than that of group, because it is not easy to construct some general examples. In this section, we will first establish several constructive theorems of HX ring and power ring, then give some examples of nontrivial HX ring and nontrivial power ring. Theorem 4.1 Let H be a subring of R and let I be a nonempty subset of R such that I+ I = I. If (h + I)(h + I) = h h + I holds for all h , h ∈ H, then H/I = {h+ I|h ∈ H} 1 2 1 2 1 2 is an HX ring on R and H/H ∩ I  H/I, where I = {x ∈ R|x+ I = I}. 2×2 Example 4.2 Let Z be the ordinary ring of 2 × 2 matrix over integral set and let ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ mn⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ H = ⎪ ⎪ |m, n ∈ Z , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I = ⎪ ⎪ |m, n ∈ Z . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 n ⎧ ⎫ ⎪ ⎪ ⎪ m n ⎪ 1 1 ⎪ ⎪ 2×2 ⎪ ⎪ Then H is a subring of Z and I + I = II = I. Let h = ⎪ ⎪ ∈ H and ⎪ ⎪ ⎩ ⎭ ⎧ ⎫ ⎪ ⎪ ⎪ m n ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ h = ⎪ ⎪ ∈ H. Then 2 ⎪ ⎪ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ h + I = ⎪ ⎪ |m, n ∈ Z , ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 n ⎧ ⎫ ⎧ ⎫ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m ⎪ ⎨ 2 ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h + I = |m , n ∈ Z . 2 ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ 0 n So, ⎧ ⎫ ⎧ ⎫ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m m m + mn ⎪ ⎨ ⎬ 1 2 1 ⎪ ⎪ ⎪ ⎪ (h + I)(h + I) = ⎪ ⎪ |m, n, m , n ∈ Z ⎪ ⎪ 1 2 ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ 0 nn Fuzzy Inf. Eng. (2009) 2: 219-228 225 ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m m n 0 m (m − n )+ mn 1 2 1 2 ⎨ 1 2 ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎪ ⎪ + ⎪ ⎪ |m, n, m , n ∈ Z . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 0 nn Considering ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m (m − n )+ mn ⎪ ⎨ ⎬ ⎪ 1 2 ⎪ ⎪ ⎪ I = ⎪ ⎪ |m, n ∈ Z, m = n , n = 1 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  ⎭ ⎩ ⎭ 0 nn ⎧ ⎫ ⎧ ⎫ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m (m − n )+ mn ⎨ 1 2 ⎬ ⎪ ⎪ ⎪ ⎪ ⊆ ⎪ ⎪ |m, n, m , n ∈ Z ⊆ I, ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ 0 nn we have that ⎧ ⎫ ⎧ ⎫ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m (m − n )+ mn ⎨ 1 2 ⎬ ⎪ ⎪ ⎪ ⎪ I = ⎪ ⎪ |m, n, m , n ∈ Z . ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ 0 nn Hence ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m m n ⎪ ⎪ m n ⎪ ⎪ m n ⎪ ⎪ 1 2 1 2 ⎪ ⎪ 1 1 ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (h + I)(h + I) = ⎪ ⎪ + I = ⎪ ⎪ ⎪ ⎪ + I 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 00 00 = h h + I. 1 2 2×2 From Theorem 4.1 we see that H/I = {h+ I|h ∈ H} is a regular HX ring on Z . Theorem 4.3 Let H be a subring of R and let I be a nonempty subset of R such that I + I = II = I. If HI = IH = 0, then H/I = {h + I|h ∈ H} is an HX ring on R and H/H ∩ I  H/I, where I = {x ∈ R|x+ I = I}. 2×2 Example 4.4 Let Z be the ordinary ring of 2× 2 matrix over integral set and let ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a 0 00 ⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H = ⎪ ⎪ |a ∈ R , I = ⎪ ⎪ |b ∈ R, b > 0 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 0 b 2×2 Then H is a subring of Z and I + I = II = I. Clearly, ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 00⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ HI = IH = ⎪ ⎪ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 2×2 so H/I is an HX ring on Z from Theorem 4.3 and it is a nonregular HX ring. Theorem 4.5 Let I be a nonempty subset of R, such that I + I = II = I. Then (I) H = {x ∈ R|xI = Ix = {0}} is a subring of R, (II) H /I = {h+ I|h ∈ H } is an HX ring on R, where H serve as an any subring 1 1 1 of H. Definition 4.6 Let H be a subring of R and let I be a subsemiring (subring) of R such that HI ∪ IH ⊆ I. Then I is called a regular semiideal (regular ideal) with respect to H. In particular, I is called a regular semiideal (regular ideal) of H if I ⊆ H. Theorem 4.7 Let H be a subring of R and let I ∈ P (R) be a regular semiideal with respect to H. Then (I) ∀h , h ∈ H, h + I ⊆ h + I, if and only if h − h ∈ I. 1 2 1 2 1 2 (II) (H/I,+,◦) is a regular power ring on R such that H/H ∩ I  H/I, where H/I = {h+ I|h ∈ H}, (h + I)◦ (h + I) = h h + I,∀h , h ∈ H. 1 2 1 2 1 2 226 Bing-xue Yao · Yu-bin Zhong (2009) Corollary 4.8 Let H be a subring of R and let I ∈ P (R) be a regular ideal with respect to H. Then (H/I,+,◦) is a uniform power ring on R. Remark The power ring (H/I,+,◦) in Corollary 4.8 is not the quotient ring of H, because I ⊆ H does not hold. 2×2 Example 4.9 Let Z be the ordinary ring of 2× 2 matrix over an integral set and let ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m⎪ ⎪ 02m⎪ ⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H = ⎪ ⎪ |m ∈ Z , I = ⎪ ⎪ |m, n ∈ Z, n  0 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 02n 2×2 2×2 Then H is a subring of Z and I is a regular semiideal of Z with respect to H. 2×2 So, (H/I,+,◦) is a regular power ring on Z from Theorem 4.7. Considering ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 04m⎪ ⎪ 00⎪ ⎪ 00⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ II = ⎪ ⎪ |m, n ∈ Z, n  0 , ⎪ ⎪ ∈ I, ⎪ ⎪  II, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 04n 02 02 we have that II  I. Hence H/I is not an HX ring. Definition 4.10 Let H be a subring of R and let I ∈ P (R) be a regular semiideal with respect to H. Then (H/I,+,◦) is called the regular quasi-quotient ring of H with respect to I. From Theorem 4.7 we can obtain the follows. Theorem 4.11 Let (R,+,◦) be a regular power ring on R. Then (I) Q is a regular semiideal with respect toR . (II) (R,+,◦) is the same as the regular quasi-quotient ring ofR with respect to Q. If I ∈ P (R) denotes a regular semiideal with respect to H , then the power ring (H/I,+,◦) is a regular power ring. To construct more general power ring, we intro- duce the following concept. Definition 4.12 Let H be a subring of R and I be a subsemiring (subring) of R such that I + I = I and HI ∪ IH ⊆ I∪{0}. Then I is called a semiideal (ideal) of H. Theorem 4.13 Let H be a subring of R and I ∈ P (R) be a semiideal with respect to H. Then (H/I,+,◦) is a power ring on R such that H/H ∩ I  H/I, where I = {x ∈ R|x+ I = I}, H/I = {h+ I|h ∈ H}, (h + I)◦ (h + I) = h h + I,∀h , h ∈ H. 1 2 1 2 1 2 Remark The conclusion of Theorem 4.13 is similar to that of Theorem 4.7, but there is a essential distinction, for the power ring in Theorem 4.7 must be a regular power ring while the power ring in Theorem 4.13 may not be regular. Definition 4.14 Let H be a subring of R and let I ∈ P (R) be a semiideal with respect to H. Then (H/I,+,◦) is called the quasi-quotient ring of H with respect to I. 3×3 Example 4.15 Let R be the ordinary ring of 3× 3 matrix over real number field Fuzzy Inf. Eng. (2009) 2: 219-228 227 and let ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a 00 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ H = ⎪ ⎪ |a ∈ R , ⎪ 000 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 00 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I = ⎪ 0 b 0 ⎪ |b ∈ R, b > 0, m ∈ Z , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 002m 3×3 Then H is a subring of R . Obviously, we have I + I = I, II ⊆ I and ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ HI = IH = ⎪ 000⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ So I is a semiideal with respect to H. Owing to that ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000  I, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 3×3 (H/I,+,◦) is a nonregular power ring on R from Theorem 4.13. Moreover, from that ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∈ I ⎪ 010 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ and ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  II, ⎪ 010 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ we have II  I. It means that H/I is not an HX ring. Acknowledgments This subject is supported by the Key Project of Chinese Ministry of Education (No. 206089). References 1. Li HX (1988) HX Ring. BUSEFAL 34:3-8 2. Zhong YB (2000) The existence of HX-ring. Applied Mathematics: A Journal of Chinese University 15(2):134-138 3. Zhong YB (1995) The structure of HX-ring on a class of ring. Fuzzy Systems and Mathematics 9(4):73-77 4. Yao BX (2001) Isomorphism theorems of regular power rings. Italian Journal of Pure and Applied Mathematics 9:91-96 228 Bing-xue Yao · Yu-bin Zhong (2009) 5. Yao BX, Li HX (2000) Power ring. Fuzzy Systems and Mathematics 14(2):15-19 6. Yao BX ( 2001) The structures of power groups. Journal of The Tripura Mathematical Society 3:1-6 7. Yao BX (2002) Generalized power group and its construction. Pure and Applied Mathematics 18(4):9-12 8. Yao BX, Li HX (2001) Weak HX-rings on a ring. Italian Journal of Pure and Applied Mathematics 10:125-131 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Upgrade of Algebraic Structure of Ring

Yao, Bing-xue; Zhong, Yu-bin
Fuzzy Information and Engineering , Volume 1 (2): 10 – Jun 1, 2009
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Taylor & Francis
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© 2009 Taylor and Francis Group, LLC
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1616-8666
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10.1007/s12543-009-0017-x
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Fuzzy Inf. Eng. (2009)2:219-228 DOI 10.1007/s12543-009-0017-x ORIGINAL ARTICLE Bing-xue Yao · Yu-bin Zhong Received: 16 January 2009/ Revised: 10 May 2009/ Accepted: 20 May 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In order to construct nonregular power rings, the concept of regular semi- ideal with respect to a subring is introduced. Then several constructive theorems of HX ring and power ring are established and some examples of non-regular power rings are constructed which contain some nonregular HX rings. Keywords HX rings · Power ring · Regular power ring · Nontrivial HX ring · Nontrivial power ring 1. Introduction With the successful upgrade of algebraic structure of group, many researchers con- sidered the upgrade of algebraic structure of some other algebraic systems, in which the ring was considered first. In 1988, Professor Li proposed the concept of HX ring [1] and derived some of their properties. Then Professor Zhong [2] gave the struc- tures of HX-ring on a class of ring and cited an example of nontrivial HX ring [3], proving the existence of nontrivial HX ring. In this paper, we establish a series of structure theorems and constructive theorems of power ring and give some examples of nontrivial HX ring and nontrivial power ring. Unless otherwise statement, (R,+,·) will always stand for an associative ring with zero element 0. 2. Concepts of HX Ring and Power Ring Let P (R) = P(R)\{Φ}, A, B ∈ P (R),we define the following operations [1]: 0 0 A+ B = {a+ b|a ∈ A, b ∈ B}, (2.1) AB = {ab|a ∈ A, b ∈ B}. (2.2) Bing-xue Yao School of Mathematics Sciences, Liaocheng University, Liaocheng 252059, P.R.China e-mail: yaobingxue@lcu.edu.cn Yu-bin Zhong () School of Mathematics and Information Sciences,Guangzhou University, Guangzhou 510006, P.R.China e-mail: Zhong yb@163.com 220 Bing-xue Yao · Yu-bin Zhong (2009) For convenience, we write a + B and aB and Ab instead of {a} + B and {a}B and A{b}, respectively. Clearly, let A, B, C ∈ P (R), then (I) A+ B = B+ A, (II) (A+ B)+ C = A+ (B+ C), (III) (AB)C = A(BC), (IV) (A(B+ C) ⊆ AB+ AC, (B+ C)A ⊆ BA+ BC, (V) A ⊆ B ⇒ A+ C ⊆ B+ C, AC ⊆ BC, CA ⊆ CB. Definition 2.1 [1] LetR be a nonempty subset of P (R) such thatR forms a ring for the operations (2.1) and (2.2). Then R is called an HX ring on R whose zero element is denoted by Q and the negative element of A∈R is denoted by−A. Example 2.2 Let S be a subring of R. Then{{S}|s ∈ S} is an HX ring on R.If A is a nonempty subset of R such that A+ A = AA = A, then {A} is also an HX ring. These HX rings are trivial HX rings. Example 2.3 [2] Let C be the set of all nonzero complex numbers. The operations ”⊕” and ”⊗”of C are defined as follows: ln|b| 0 a⊕ b = ab, a⊗ b = |a| ,∀a, b ∈ C . Then (C ,⊕,⊗) forms a ring. Let I = (1,+∞) and H = {1,−1, i,−i}. ThenR = {a⊕ I|a ∈ H is an HX ring on(C ,⊕,⊗). Remark A quotient ring of a given ring R,may not be an HX ring, because the multiplication of the quotient ring is not coincident with the operation (2.2), although its addition coincides with the operation (2.1). Example 2.4 [4] Let Z denote the ordinary integral number ring and letI =< 10 > be the ideal generated by 10. Then the quotient ring (Z/I,+,◦)of Z with respect to I satisfies: (z + I)+ (z + I) = (z + z )+ I, (z + I)◦ (z + I) = z z + I,∀z , z ∈ Z. 1 2 1 2 1 2 1 2 1 2 Let z = 2, z = 4. Then 1 2 (z + I)(z + I) = {8+ 20m+ 40n+ 100mn|m, n ∈ Z}, 1 2 z z + I = {8+ 10m|m ∈ Z}. 1 2 Clearly, (z + I)◦ (z + I) = z z + I  (z + I)(z + I), 1 2 1 2 1 2 since 18 ∈ z z + I and 18  (z + I)(z + I). This means that (Z/I,+,◦) is not an HX 1 2 1 2 ring. In fact, if I is an ideal of R, then for all r , r ∈ R we have 1 2 (r + I)(r + I) ⊆ r r + I. 1 2 1 2 However, in general, “⊆” can not be replaced by “=”. Definition 2.5 [5] Let R be a nonempty subset of P (R) with two operations, ad- dition “+” and multiplication “◦”, such that A + B = {a + b|a ∈ A, b ∈ B} and Fuzzy Inf. Eng. (2009) 2: 219-228 221 A◦ B ⊇ AB for all A, B∈R.If (R,+,◦) forms a ring, then (R,+,◦) is called a power ring on R whose zero element is denoted by Q. Clearly, (I) if (R,+,◦) is a power ring on R, then (R,+) is a power group on (R,+); (II) all quotient rings of every subring of R and all HX rings on R are also power rings on R. Example 2.6 Let R be the ordinary real number ring and let E = (0,+∞),R = {a+ E|a ∈ R}. We define operations in R as follows: (a+ E)+ (b+ E) = (a+ b)+ E, (a+ E)⊗ (b+ E) = ab+ E,∀a, b ∈ R. Then (R,+,⊗) is a ring but not a power ring because of (−1+ E)⊗ (2+ E) = −2+ E and −2+ E  (−1+ E)(2+ E). Definition 2.7 All quotient rings of every subring of R and all trivial HX rings on R are called trivial power rings on R. 3. Structure of Power Ring Let (R,+,◦) be a power ring on R. Then R = ∪{A|A ∈R} is called a basic element set of (R,+,◦). Definition 3.1 Let S be a nonempty subset of R such that S + S ⊆ S and S S ⊆ S .Then S is called a subsemiring of R. Theorem 3.2 Let (R,+,◦) be a power ring on R. Then (I) A ⊆ B⇒−B⊆−A,∀A, B∈R, (II) AQ∪ QA ⊆ Q,∀A∈R, (III) R is a subsemiring of R, (IV) Q is a subsemiring of R. Theorem 3.3 Let (R,+,◦) be a power ring on R. Then 0∈R ⇒ 0 ∈ Q. Theorem 3.4 Let (R,+,◦) be a power ring on R. Then (I) A ⊆ B ⇒ AB = BA = AA = BB,∀A, B∈R, (II) Q ⊆ A ⇒ AB = BA = Q,∀B∈R. Definition 3.5 Let (R,+,◦) be a power ring on R and let A ∈R. Then A = {a ∈ A|(−a) ∈ (−A)} is called the kernel of A. Obviously, the concept of kernel coincides with that of power group [6,7]. Definition 3.6 Let (R,+,◦) be a power ring on R. If 0 ∈ Q. Then (R,+,◦) is called a regular power ring; if Q is a subring of R, then (R,+,◦) is called a uniform power ring. In particular, if R is an HX ring on R, we have the concepts of regular HX ring and uniform HX ring. If (R,+,◦) is a regular (uniform) power ring, then we say that (R,+,◦) is regular (uniform). Clearly, if (R,+,◦) is a regular (uniform) power ring on R, then (R,+) is a regular (uniform) power group on (R,+), so we have the following conclusions similar to those of a power group. Theorem 3.7 Let (R,+,◦) be a regular power ring on R. Then A  Φ for all A∈R. 222 Bing-xue Yao · Yu-bin Zhong (2009) Theorem 3.8 Let (R,+,◦) be a power ring on R. If there exists A ∈R such that A  Φ, then (R,+,◦) is a regular power ring. Theorem 3.9 Let (R,+,◦) be a regular power ring on R. Then for all A, B∈R,we have (I) A+ B = A+ B, (II) AB∪ AB ⊆ A◦ B. Theorem 3.10 Let (R,+,◦) be a regular power group on R. Then for all A, B∈R, we have (I) QA∪ AQ ⊆ Q, (II) A B ⊆ A◦ B. In general, “A B = A◦ B ” does not hold even if in a regular power ring. 3×3 Example 3.11 Let R be an ordinary ring of 3× 3 matrix over a real number field and let ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a 00⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ H = ⎪ 000⎪ |a ∈ R , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I = ⎪ ⎪ |b, c, x ∈ R, b  0, c  0 . ⎪ 0 bx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 00 c Then ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I + I = I, HI = IH = ⎪ 000⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ Let b , b , c , c , x , x ∈ R such that b , b , c , c  0. Then 1 2 1 2 1 2 1 2 1 2 ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 00 0 ⎪ ⎪ 00 0 ⎪ ⎪ 00 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 b x ⎪ ⎪ 0 b x ⎪ = ⎪ 0 b b b x + c x ⎪ ∈ I, ⎪ 1 1 ⎪ ⎪ 2 2 ⎪ ⎪ 1 2 1 2 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 c 00 c 00 c c 1 2 1 2 that is, II ⊆ I. Let b, c, x ∈ R such that b, c  0. Then ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ 000⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 bx⎪ = ⎪ 0 bx⎪ ⎪ 010⎪ ∈ II, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 c 00 c 001 that is, I ⊆ II. So, II = I. 3×3 Now we show that {h+ I|h ∈ H} is a nontrivial HX ring on R . Let h , h ∈ H. Then 1 2 (h + I)(h + I) ⊆ h h + h I + Ih + II = h h + I. 1 2 1 2 1 2 1 2 Conversely, if x ∈ I, then there exist x , x ∈ I, such that x = x x . Sowehave 1 2 1 2 h h + x = h h + x x = (h + x )(h + x ) ∈ (h + I)(h + I). 1 2 1 2 1 2 1 1 2 2 1 2 Fuzzy Inf. Eng. (2009) 2: 219-228 223 That is, h h + I ⊆ (h + I)(h + I). Thus we have 1 2 1 2 (h + I)(h + I) = h h + I. 1 2 1 2 3×3 Hence {h+ I|h ∈ H} is an HX ring on R . Moreover, {h+ I|h ∈ H} is a regular HX 3×3 ring because of I contains the zero element of R . It is obvious that ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ II = I = ⎪ 00 x⎪ |x ∈ R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ and ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I I = ⎪ 000⎪ , soI I  II. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ Theorem 3.12 Let (R,+,◦) be a regular power ring on R. Then R = ∪{A|A∈R} is a subring of R and Q is an ideal of R . Theorem 3.13 Let (R,+,◦) be a regular power ring on R. Then the following con- ditions are equivalent. (I) a ∈ A, (II) A = a+ Q, (III) (−A) = (−a)+ Q, (IV) A = a+ Q. Theorem 3.14 Let (R,+,◦) be a regular power ring on R. Then (I) A, B∈R⇒ A+ B = (a+ Q)+ (b+ Q) = (a+ b)+ Q, where a ∈ A, b ∈ B, (II) A, B∈R⇒ A◦ B = (a+ Q)◦ (b+ Q) = ab+ Q, where a ∈ A, b ∈ B, (III) R = {a+ Q|a ∈ R }, (IV) R /Q  (R,+,◦). Proof We only prove (II) and (IV). (II) From a ∈ A and b ∈ B we have ab ∈ A B ⊆ A◦ B, so A◦ B = ab+ Q, that is, A◦ B = (a+ Q)◦ (b+ Q) = ab+ Q. (IV) Clearly,R is a subring of R from Theorem 3.12. Let ∗ ∗ f : R →R, a → f (a) = a+ Q,∀a ∈ R . Then f is an epimorphism and ∗ ∗ ker f = {a ∈ R |a+ Q = Q} = {a ∈ R |a ∈ Q,−a ∈ Q} = Q. Hence R /Q  (R,+,◦). 224 Bing-xue Yao · Yu-bin Zhong (2009) The above theorem shows that the multiplication of a regular power ring is de- termined even though the multiplication of a power ring is not defined specifically. Later, we will show that a regular power ring is isomorphic to a quotient ring. Theorem 3.15 Let (R,+,◦) be a uniform power ring on R and let A∈R. Then (I) A = A, − − (II) −A = A , where A = {−a|a ∈ A}. Theorem 3.16 Let (R,+,◦) be a power ring on R. If there exists A ∈R, such that A = Aor−A = A , then (R,+,◦) is a uniform power ring. Theorem 3.17 Let (R,+,◦) be a uniform power ring on R. Then (I) R is a subring of R, (II) Q is an ideal of R , (III) A∈R⇒ A = a+ Q, where a ∈ A, ∗ ∗ ∗ (IV) (R,+,◦) = R /Q, where R /Q is the quotient ring of R with respect to Q. 4. Construction of HX Ring and Power Ring Upgrade of algebraic structure of ring is more difficult than that of group, because it is not easy to construct some general examples. In this section, we will first establish several constructive theorems of HX ring and power ring, then give some examples of nontrivial HX ring and nontrivial power ring. Theorem 4.1 Let H be a subring of R and let I be a nonempty subset of R such that I+ I = I. If (h + I)(h + I) = h h + I holds for all h , h ∈ H, then H/I = {h+ I|h ∈ H} 1 2 1 2 1 2 is an HX ring on R and H/H ∩ I  H/I, where I = {x ∈ R|x+ I = I}. 2×2 Example 4.2 Let Z be the ordinary ring of 2 × 2 matrix over integral set and let ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ mn⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ H = ⎪ ⎪ |m, n ∈ Z , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I = ⎪ ⎪ |m, n ∈ Z . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 n ⎧ ⎫ ⎪ ⎪ ⎪ m n ⎪ 1 1 ⎪ ⎪ 2×2 ⎪ ⎪ Then H is a subring of Z and I + I = II = I. Let h = ⎪ ⎪ ∈ H and ⎪ ⎪ ⎩ ⎭ ⎧ ⎫ ⎪ ⎪ ⎪ m n ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ h = ⎪ ⎪ ∈ H. Then 2 ⎪ ⎪ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ h + I = ⎪ ⎪ |m, n ∈ Z , ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 n ⎧ ⎫ ⎧ ⎫ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m ⎪ ⎨ 2 ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h + I = |m , n ∈ Z . 2 ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ 0 n So, ⎧ ⎫ ⎧ ⎫ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m m m + mn ⎪ ⎨ ⎬ 1 2 1 ⎪ ⎪ ⎪ ⎪ (h + I)(h + I) = ⎪ ⎪ |m, n, m , n ∈ Z ⎪ ⎪ 1 2 ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ 0 nn Fuzzy Inf. Eng. (2009) 2: 219-228 225 ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m m n 0 m (m − n )+ mn 1 2 1 2 ⎨ 1 2 ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎪ ⎪ + ⎪ ⎪ |m, n, m , n ∈ Z . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 0 nn Considering ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m (m − n )+ mn ⎪ ⎨ ⎬ ⎪ 1 2 ⎪ ⎪ ⎪ I = ⎪ ⎪ |m, n ∈ Z, m = n , n = 1 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  ⎭ ⎩ ⎭ 0 nn ⎧ ⎫ ⎧ ⎫ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m (m − n )+ mn ⎨ 1 2 ⎬ ⎪ ⎪ ⎪ ⎪ ⊆ ⎪ ⎪ |m, n, m , n ∈ Z ⊆ I, ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ 0 nn we have that ⎧ ⎫ ⎧ ⎫ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m (m − n )+ mn ⎨ 1 2 ⎬ ⎪ ⎪ ⎪ ⎪ I = ⎪ ⎪ |m, n, m , n ∈ Z . ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎩ ⎭ ⎩ ⎭ 0 nn Hence ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m m m n ⎪ ⎪ m n ⎪ ⎪ m n ⎪ ⎪ 1 2 1 2 ⎪ ⎪ 1 1 ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (h + I)(h + I) = ⎪ ⎪ + I = ⎪ ⎪ ⎪ ⎪ + I 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 00 00 = h h + I. 1 2 2×2 From Theorem 4.1 we see that H/I = {h+ I|h ∈ H} is a regular HX ring on Z . Theorem 4.3 Let H be a subring of R and let I be a nonempty subset of R such that I + I = II = I. If HI = IH = 0, then H/I = {h + I|h ∈ H} is an HX ring on R and H/H ∩ I  H/I, where I = {x ∈ R|x+ I = I}. 2×2 Example 4.4 Let Z be the ordinary ring of 2× 2 matrix over integral set and let ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a 0 00 ⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H = ⎪ ⎪ |a ∈ R , I = ⎪ ⎪ |b ∈ R, b > 0 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 0 b 2×2 Then H is a subring of Z and I + I = II = I. Clearly, ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 00⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ HI = IH = ⎪ ⎪ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 2×2 so H/I is an HX ring on Z from Theorem 4.3 and it is a nonregular HX ring. Theorem 4.5 Let I be a nonempty subset of R, such that I + I = II = I. Then (I) H = {x ∈ R|xI = Ix = {0}} is a subring of R, (II) H /I = {h+ I|h ∈ H } is an HX ring on R, where H serve as an any subring 1 1 1 of H. Definition 4.6 Let H be a subring of R and let I be a subsemiring (subring) of R such that HI ∪ IH ⊆ I. Then I is called a regular semiideal (regular ideal) with respect to H. In particular, I is called a regular semiideal (regular ideal) of H if I ⊆ H. Theorem 4.7 Let H be a subring of R and let I ∈ P (R) be a regular semiideal with respect to H. Then (I) ∀h , h ∈ H, h + I ⊆ h + I, if and only if h − h ∈ I. 1 2 1 2 1 2 (II) (H/I,+,◦) is a regular power ring on R such that H/H ∩ I  H/I, where H/I = {h+ I|h ∈ H}, (h + I)◦ (h + I) = h h + I,∀h , h ∈ H. 1 2 1 2 1 2 226 Bing-xue Yao · Yu-bin Zhong (2009) Corollary 4.8 Let H be a subring of R and let I ∈ P (R) be a regular ideal with respect to H. Then (H/I,+,◦) is a uniform power ring on R. Remark The power ring (H/I,+,◦) in Corollary 4.8 is not the quotient ring of H, because I ⊆ H does not hold. 2×2 Example 4.9 Let Z be the ordinary ring of 2× 2 matrix over an integral set and let ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 m⎪ ⎪ 02m⎪ ⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H = ⎪ ⎪ |m ∈ Z , I = ⎪ ⎪ |m, n ∈ Z, n  0 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 00 02n 2×2 2×2 Then H is a subring of Z and I is a regular semiideal of Z with respect to H. 2×2 So, (H/I,+,◦) is a regular power ring on Z from Theorem 4.7. Considering ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 04m⎪ ⎪ 00⎪ ⎪ 00⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ II = ⎪ ⎪ |m, n ∈ Z, n  0 , ⎪ ⎪ ∈ I, ⎪ ⎪  II, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 04n 02 02 we have that II  I. Hence H/I is not an HX ring. Definition 4.10 Let H be a subring of R and let I ∈ P (R) be a regular semiideal with respect to H. Then (H/I,+,◦) is called the regular quasi-quotient ring of H with respect to I. From Theorem 4.7 we can obtain the follows. Theorem 4.11 Let (R,+,◦) be a regular power ring on R. Then (I) Q is a regular semiideal with respect toR . (II) (R,+,◦) is the same as the regular quasi-quotient ring ofR with respect to Q. If I ∈ P (R) denotes a regular semiideal with respect to H , then the power ring (H/I,+,◦) is a regular power ring. To construct more general power ring, we intro- duce the following concept. Definition 4.12 Let H be a subring of R and I be a subsemiring (subring) of R such that I + I = I and HI ∪ IH ⊆ I∪{0}. Then I is called a semiideal (ideal) of H. Theorem 4.13 Let H be a subring of R and I ∈ P (R) be a semiideal with respect to H. Then (H/I,+,◦) is a power ring on R such that H/H ∩ I  H/I, where I = {x ∈ R|x+ I = I}, H/I = {h+ I|h ∈ H}, (h + I)◦ (h + I) = h h + I,∀h , h ∈ H. 1 2 1 2 1 2 Remark The conclusion of Theorem 4.13 is similar to that of Theorem 4.7, but there is a essential distinction, for the power ring in Theorem 4.7 must be a regular power ring while the power ring in Theorem 4.13 may not be regular. Definition 4.14 Let H be a subring of R and let I ∈ P (R) be a semiideal with respect to H. Then (H/I,+,◦) is called the quasi-quotient ring of H with respect to I. 3×3 Example 4.15 Let R be the ordinary ring of 3× 3 matrix over real number field Fuzzy Inf. Eng. (2009) 2: 219-228 227 and let ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a 00 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ H = ⎪ ⎪ |a ∈ R , ⎪ 000 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 00 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ I = ⎪ 0 b 0 ⎪ |b ∈ R, b > 0, m ∈ Z , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 002m 3×3 Then H is a subring of R . Obviously, we have I + I = I, II ⊆ I and ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ HI = IH = ⎪ 000⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ So I is a semiideal with respect to H. Owing to that ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000  I, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 3×3 (H/I,+,◦) is a nonregular power ring on R from Theorem 4.13. Moreover, from that ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∈ I ⎪ 010 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ and ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 000⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  II, ⎪ 010 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ we have II  I. It means that H/I is not an HX ring. Acknowledgments This subject is supported by the Key Project of Chinese Ministry of Education (No. 206089). References 1. Li HX (1988) HX Ring. BUSEFAL 34:3-8 2. Zhong YB (2000) The existence of HX-ring. Applied Mathematics: A Journal of Chinese University 15(2):134-138 3. Zhong YB (1995) The structure of HX-ring on a class of ring. Fuzzy Systems and Mathematics 9(4):73-77 4. Yao BX (2001) Isomorphism theorems of regular power rings. Italian Journal of Pure and Applied Mathematics 9:91-96 228 Bing-xue Yao · Yu-bin Zhong (2009) 5. Yao BX, Li HX (2000) Power ring. Fuzzy Systems and Mathematics 14(2):15-19 6. Yao BX ( 2001) The structures of power groups. Journal of The Tripura Mathematical Society 3:1-6 7. Yao BX (2002) Generalized power group and its construction. Pure and Applied Mathematics 18(4):9-12 8. Yao BX, Li HX (2001) Weak HX-rings on a ring. Italian Journal of Pure and Applied Mathematics 10:125-131

Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Jun 1, 2009

Keywords: HX rings; Power ring; Regular power ring; Nontrivial HX ring; Nontrivial power ring

References

  • HX Ring
  • The existence of HX-ring
  • The structure of HX-ring on a class of ring
  • Isomorphism theorems of regular power rings
  • Power ring
  • The structures of power groups
  • Generalized power group and its construction
  • Weak HX-rings on a ring