$I$-rings

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.