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.


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