## 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)

### MSC:

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

### Citations:

Zbl 0840.16006; Zbl 0839.16004; Zbl 0768.16001
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### References:

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