## Rings for which certain elements have the principal extension property.(English)Zbl 1040.16004

Throughout all rings $$R$$ are associative with identity and the modules are right unitary $$R$$-modules. For a module $$M$$, $$J(M)$$ denotes the Jacobson radical. The left and right annihilator operators are denoted by $${\mathbf l}(\;)$$ and $${\mathbf r}(\;)$$. The right socle and the right singular ideal of $$R$$ are denoted by $$S_r$$ and $$Z_r$$, respectively. For a module $$M$$, put $$p(M)=\{a\in R\mid{\mathbf l}_M({\mathbf r}_R(a))=Ma\}$$ and $$gp(M)=\{a\in R\mid\exists n>0$$ such that $$a^n\neq 0$$ and $${\mathbf l}_M({\mathbf r}_R(a^n))=Ma^n\}\cup\{0\}$$. A module $$M$$ is called P-injective if $$p(M)=R$$ and $$M$$ is said to satisfy GC2 if for any $$N\subseteq M$$ with $$N\cong M$$, $$N$$ is a summand of $$M$$. A ring $$R$$ is called: (i) semiregular if $$R/J(R)$$ is regular and idempotents of $$R/J(R)$$ lift to idempotents of $$R$$; (ii) right JP-injective if $$J(R)\subseteq p(R_R)$$; (iii) right JGP-injective if $$J(R)\subseteq gp(R_R)$$. A subset $$X$$ of $$R$$ is said to be essential in $$R_R$$ if for any $$0\neq a\in R$$, there exists $$b\in R$$ such that $$0\neq ab\in X$$.
The main results established by the authors are: (1) Let $$R$$ be a right CS-ring satisfying GC2. (i) If $$R$$ has ACC on essential right ideals, then $$R$$ is right Artinian. (ii) If $$R$$ has ACC on right annihilators, then $$R$$ is semiprimary. (iii) If $$R/S_r$$ has ACC on right annihilators, then $$R$$ is semilocal and $$Z_r$$ is nilpotent; (2) If $$R$$ is right JGP-injective, then $$J(R)\subseteq Z_r$$; (3) If $$I$$ is an ideal such that $$R/I$$ has ACC on right annihilators of principal left ideals and $$R$$ is a right JGP-injective ring, then $${\mathbf l}(I)\cap J(R)$$ is right T-nilpotent; (4) Every semiregular right JP-injective ring $$R$$ is right P-injective; (5) If $$gp(R_R)$$ is essential in $$R_R$$, and for any infinite sequence of elements $$x_1,x_2,x_3,\dots$$ in $$R$$ the chain $${\mathbf r}(x_1)\subseteq{\mathbf r}(x_2x_1)\subseteq{\mathbf r}(x_3x_2x_1)\subseteq\cdots$$ terminates, then $$J(R)=Z_r$$ and $$R$$ is right perfect.

### MSC:

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