# On the lower radical construction

On the lower radical construction Acta Mathematiea Academiae Seientiarum Hungaricae Tomus 25 (1--2), (1974), pp. 31--32. LOWER RADICAL~CONSTRUCTION ON THE By P. N. STEWART (Adelaide) Let J{ be a non-empty, homomorphically closed class of zero (R z =(0)) rings. Put JCl 1 =rig, and for integers i>1, define dd 1 to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal in J/{~_~. In this way we obtain an ascending chain d//~/~e___ .... the union of which is the smallest radical class containing J/l . We shall denote this radical class by L(dg), it is the lower radical class determined by all. GARDNER proves that L(d//)----dl 4 , and in  an example is given of a class J/{ of zero rings such that L(~)~ Jr 2 . In this note we prove: L(JZ) =~3. PROOF. 1'~ Let k be an integer _->4 and R be a ringin Jr There is a chain of sub- rings A= As CA2 c C Ak= R such that A~ is an ideal of Ai+l for l<-_i<=k-1, and A~Cd//i for l<=i<=k . Let A=A +AR+RA +RAR be the ideal of R which is generated by A. It is http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

# On the lower radical construction

, Volume 25 (2) – Jun 18, 2005
2 pages      Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01901743
Publisher site
See Article on Publisher Site

### Abstract

Acta Mathematiea Academiae Seientiarum Hungaricae Tomus 25 (1--2), (1974), pp. 31--32. LOWER RADICAL~CONSTRUCTION ON THE By P. N. STEWART (Adelaide) Let J{ be a non-empty, homomorphically closed class of zero (R z =(0)) rings. Put JCl 1 =rig, and for integers i>1, define dd 1 to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal in J/{~_~. In this way we obtain an ascending chain d//~/~e___ .... the union of which is the smallest radical class containing J/l . We shall denote this radical class by L(dg), it is the lower radical class determined by all. GARDNER proves that L(d//)----dl 4 , and in  an example is given of a class J/{ of zero rings such that L(~)~ Jr 2 . In this note we prove: L(JZ) =~3. PROOF. 1'~ Let k be an integer _->4 and R be a ringin Jr There is a chain of sub- rings A= As CA2 c C Ak= R such that A~ is an ideal of Ai+l for l<-_i<=k-1, and A~Cd//i for l<=i<=k . Let A=A +AR+RA +RAR be the ideal of R which is generated by A. It is

### Journal

Acta Mathematica Academiae Scientiarum HungaricaSpringer Journals

Published: Jun 18, 2005

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