Generalizations of principally injective rings. (English) Zbl 0923.16002

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)


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


[1] Anderson, F. W.; Fuller, K. R., Rings and Categories of Modules (1974), Springer-Verlag: Springer-Verlag New York · Zbl 0242.16025
[2] Armendariz, E. P.; Park, J. K., Self-injective rings with restricted chain conditions, Arch. Math., 58, 24-33 (1992) · Zbl 0768.16001
[3] Camillo, V., Commutative rings whose principal ideals are annihilators, Protugal. Math., 46, 33-37 (1989) · Zbl 0668.13007
[4] Fisher, J. W., On the nilpotency of nil subrings, Canad. J. Math., 22, 1211-1216 (1970) · Zbl 0194.34602
[5] Jain, S. K.; López-Permouth, S. R., Rings whose cyclics are essentially embeddable in projectives, J. Algebra, 128, 257-269 (1990) · Zbl 0698.16012
[6] Koh, K., On some characteristic properties of self-injective rings, Proc. Amer. Math. Soc., 19, 209-213 (1968) · Zbl 0155.07205
[7] Nam, S. B.; Kim, N. K.; Kim, J. Y., On simple GP-injective modules, Comm. Algebra, 14, 5437-5444 (1995) · Zbl 0840.16006
[8] Mohamed, S. H.; Müller, B. J., Continuous and Discrete Modules (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0701.16001
[9] Nicholson, W. K.; Yousif, M. F., Principally injective rings, J. Algebra, 174, 77-93 (1995) · Zbl 0839.16004
[10] Nicholson, W. K.; Yousif, M. F., On a theorem of Camillo, Comm. Algebra, 14, 5309-5314 (1995) · Zbl 0839.16005
[11] Yousif, M. F., On semiperfect FPF-rings, Canad. Math. Bull., 37, 287-288 (1994) · Zbl 0816.16004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.