Generalizations of principally injective rings. (English) Zbl 0923.16002

Page, Stanley S.; Zhou, Yiqiang

J. Algebra 206, No. 2, 706-721 (1998).

Given a module \(M_R\), denote \(S=\text{End}_R(M)\). The module \(M\) is said to be almost principally injective (or simply AP-injective) if, for any \(a\in R\), there is an \(S\)-submodule \(X_a\) of \(M\) such that \(\ell_M(r_R(a))=a\oplus X_a\) as left \(S\)-modules. The module \(M\) is called almost general principally injective (or simply AGP-injective) if, for any \(0\neq a\in R\), there is a positive integer \(n(a)\), depending on \(a\), and an \(S\)-submodule \(X_a\) of \(M\) such that \(a^n\neq 0\) and \(\ell_M(r_R(a))=Ma^n\oplus X_a\) as left \(S\)-modules. If \(R_R\) is an AP-injective (resp. AGP-injective) module, we call \(R\) a right AP-injective (resp. right AGP-injective) ring. Furthermore, a ring \(R\) is called a right QP-injective ring if, for any \(0\neq a\in R\), there is a left ideal \(X_a\) such that \(\ell_R(r_R(a))=Ra+X_a\) with \(a\not\in X_a\). More generally, a ring \(R\) is called right QGP-injective if, for each \(0\neq a\in R\), there is a positive integer \(n=n(a)\) depending on \(a\) such that \(\ell_R(r_R(a))=Ra^n+X_a\) such that \(a^n\not\in X_a\).
After providing several interesting examples which show the AP-injectivity and the GP-injectivity are distinct, the relationship between the Jacobson radical \(J(R)\) and the right singular ideal \(Z(R_R)\) is investigated. For example, it is shown that if \(R\) is a right AGP-injective ring, then \(J(R)=Z(R_R)\). Thereby results of S. B. Nam, N. K. Kim and J. Y. Kim [Commun. Algebra 23, No. 14, 5437-5444 (1995; Zbl 0840.16006)] and W. K. Nicholson and M. F. Yousif [J. Algebra 174, No. 1, 77-93 (1995; Zbl 0839.16004)] can be extended. Also it is proved that if \(R\) is semiperfect right QGP-injective, then \(\text{Soc}(R_R)\subseteq\text{Soc}({_RR})\) and \(Z({_RR})\subseteq Z(R_R)=J(R)\).
For the nilpotency of \(J(R)\), it is proved that \(J(R)\) is nilpotent if \(R\) is right AGP-injective and \(R/\text{Soc}(R_R)\) satisfies the ACC on right annihilators. This result extends a result of E. P. Armendariz and J. K. Park [Arch. Math. 58, No. 1, 24-33 (1992; Zbl 0768.16001)].
A decomposition of a right QGP-injective ring with an additional assumption into a semisimple ring and a ring with square zero right socle is investigated. Also necessary and sufficient conditions for a matrix ring to be an AP-injective ring are considered.

Reviewer: J.K.Park (Pusan)

MSC:

16D50	Injective modules, self-injective associative rings
16P60	Chain conditions on annihilators and summands: Goldie-type conditions

Keywords:

AP-injectivity; GP-injectivity; Jacobson radical; right singular ideals; right AGP-injective rings; ACC on right annihilators; right QGP-injective rings; right socles; AP-injective rings

Citations:

Zbl 0840.16006; Zbl 0839.16004; Zbl 0768.16001

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