Minimal generators of radical classes

Leavitt, W. G.; Rossa, R. F.

doi:10.1080/00927878608823324

Cited by 2 publications

(4 citation statements)

References 3 publications

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“…Since K has a unit pA has an element pe which which is a non-zero idempotent modulo H. But then pe --(pe) 2 E H and since p2e2 ~ H, this is a contradiction. By the same proof as in [1] we also have COROLLARY 8. If R has a unit and satisfies a one variable polynomial identity then…”

Section: A ~ R ~ R/pr ----Kmentioning

confidence: 65%

“…As we noted in [1], since the Jacobson radical contains the ideal (p) of Z(p), it follows that the Jacobson radical does not have a minimal generator.…”

mentioning

confidence: 91%

“…

In a recent paper [1] we defined, for a ring R, the class

T~EO~M 1 ([1], Theorem 4). An HH class M is a minimal generator for the radical P = LM if and only if F(R) fq M ~ 0 for all R ~ M,

and TH~OI~M 2 ([1], Theorem 6).

…”

mentioning

confidence: 99%

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A note on minimal generators of hereditary radicals

Leavitt

1988

Period Math Hung

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In a recent paper [1] we defined, for a ring R, the class T~EO~M 1 ([1], Theorem 4). An HH class M is a minimal generator for the radical P = LM if and only if F(R) fq M ~ 0 for all R ~ M, and TH~OI~M 2 ([1], Theorem 6). For a non-zero HH class M, the radical P ~-LM has a minimal generator if and only if every 0 :/= R E M has an ideal 0 ~ I which is either nilpotent or is a ring for which F(I) f3 M ~ O.The following result contains all known cases of hereditary radicals with minimal generators: THEOREM 3. Let M be an HH class. Then P ~--LM has a minimal generator if/or each R E M either:

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Section: A ~ R ~ R/pr ----Kmentioning

confidence: 65%

“…As we noted in [1], since the Jacobson radical contains the ideal (p) of Z(p), it follows that the Jacobson radical does not have a minimal generator.…”

mentioning

confidence: 91%

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A note on minimal generators of hereditary radicals

Leavitt

1988

Period Math Hung

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“…It follows from the construction of the intersection radical (EJ = E  J) of a ring A (see Leavitt [2]) that A is EJ-semisimple if and only if E(A)  J(A) = 0.…”

Section: ) Ej-semisimple Ringsmentioning

confidence: 99%

Three Distinct Non-Hereditary Radicals Which Coincide with the Classical Radical for Rings with D.C.C.

Majumdar

Dey

2016

GANIT: J. Bangladesh Math. Soc.

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Majumdar and Paul [3] introduced and studied a new radical E defined as the upper radical determined by the class of all rings each of whose ideals is idempotent. In this paper the authors continue the study further and also study the join radical and the intersection radical (due to Leavitt) obtained from E and the Jacobson radical J. These have been denoted by E + J and EJ respectively. The radical and the semisimple rings corresponding to E + J and EJ have been obtained. Both of these radicals coincide with the classical nil radical for Artinian rings. Important properties of these radicals and their position among the well-known special radicals have been investigated. It has been proved that E, EJ and E + J are non-hereditary. It has also been proved as an independent result that the nil radical N is not dual, i.e., N ≠ N  .

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Minimal generators of radical classes

Cited by 2 publications

References 3 publications

A note on minimal generators of hereditary radicals

A note on minimal generators of hereditary radicals

Three Distinct Non-Hereditary Radicals Which Coincide with the Classical Radical for Rings with D.C.C.

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