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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ I$-rings

Author: W. K. Nicholson
Journal: Trans. Amer. Math. Soc. 207 (1975), 361-373
MSC: Primary 16A32
MathSciNet review: 0379576
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Abstract: A ring $ R$, possibly with no identity, is called an $ {I_0}$-ring if each one-sided ideal not contained in the Jacobson radical $ J(R)$ contains a nonzero idempotent. If, in addition, idempotents can be lifted modulo $ J(R),R$ is called an $ I$-ring. A survey of when these properties are inherited by related rings is given. Maximal idempotents are examined and conditions when $ {I_0}$-rings have an identity are given. It is shown that, in an $ {I_0}$-ring $ R$, primitive idempotents are local and primitive idempotents in $ R/J(R)$ can always be lifted. This yields some characterizations of $ {I_0}$-rings $ R$ such that $ R/J(R)$ is primitive with nonzero socle. A ring $ R$ (possibly with no identity) is called semiperfect if $ R/J(R)$ is semisimple artinian and idempotents can be lifted modulo $ J(R)$. These rings are characterized in several new ways: among them as $ {I_0}$-rings with no infinite orthogonal family of idempotents, and as $ {I_0}$-rings $ R$ with $ R/J(R)$ semisimple artinian. Several other properties are derived. The connection between $ {I_0}$-rings and the notion of a regular module is explored. The rings $ R$ which have a regular module $ M$ such that $ J(R) = \operatorname{ann} (M)$ are studied. In particular they are $ {I_0}$-rings. In addition, it is shown that, over an $ {I_0}$-ring, the endomorphism ring of a regular module is an $ {I_0}$-ring with zero radical.

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Keywords: $ I$-ring, idempotents, lifting idempotents, semiperfect ring, von Neumann regular ring, regular module, endomorphism ring
Article copyright: © Copyright 1975 American Mathematical Society