𝐼-rings

Abstract

A ring R R , possibly with no identity, is called an I 0 {I_0} -ring if each one-sided ideal not contained in the Jacobson radical J ( R ) J(R) contains a nonzero idempotent. If, in addition, idempotents can be lifted modulo J ( R ) , R J(R),R is called an I I -ring. A survey of when these properties are inherited by related rings is given. Maximal idempotents are examined and conditions when I 0 {I_0} -rings have an identity are given. It is shown that, in an I 0 {I_0} -ring R R , primitive idempotents are local and primitive idempotents in R / J ( R ) R/J(R) can always be lifted. This yields some characterizations of I 0 {I_0} -rings R R such that R / J ( R ) R/J(R) is primitive with nonzero socle. A ring R R (possibly with no identity) is called semiperfect if R / J ( R ) R/J(R) is semisimple artinian and idempotents can be lifted modulo J ( R ) J(R) . These rings are characterized in several new ways: among them as I 0 {I_0} -rings with no infinite orthogonal family of idempotents, and as I 0 {I_0} -rings R R with R / J ( R ) R/J(R) semisimple artinian. Several other properties are derived. The connection between I 0 {I_0} -rings and the notion of a regular module is explored. The rings R R which have a regular module M M such that J ( R ) = ann ⁡ ( M ) J(R) = \operatorname {ann} (M) are studied. In particular they are I 0 {I_0} -rings. In addition, it is shown that, over an I 0 {I_0} -ring, the endomorphism ring of a regular module is an I 0 {I_0} -ring with zero radical.