Soc-Injective Rings and Modules*

I-n-Coherent Rings, I-n-Semihereditary Rings, and I-Regular Rings

“…Let M be any I-n-flat right R-module. Then M + is I-n-injective left R-module by Theorem 3.1 and, hence, M + is n-injective by (2). Thus, M is n-flat by Corollary 3.1.…”

“…A left R-module M is n-injective if and only if M is R-n-injective and a left R-module M is J-injective if and only if M is J-n-injective for every positive integer n. Following [2], a ring R is said to be left Soc-injective if every R-homomorphism from a semisimple submodule of R R to R extends to R. Clearly, if Soc( R R) is finitely generated, then R is left Soc-injective if and only if R R is Soc( R R)-n-injective for every positive integer n. We note that J-P-injective modules are called JP -injective in [22]. (1) M is I-n-injective.…”

“…Assume (2). Then for every injective right R-module E, E has a monic I-n-flat preenvelope F, then E is isomorphism to a direct summand of F and, thus, E is I-n-flat.…”

Rad-injective and Almost-injective Modules and Rings

“…M is called strongly soc-injective if M is soc-N -injective for every right R-module N , or equivalently, if M = E ⊕ K, where E is injective with essential socle and soc(K) = 0 (see [1]). Thus, the Z-module Z 6 is almost-injective but not strongly soc-injective.…”

ASPECTS OF WEAK, s-CS AND ALMOST INJECTIVE RINGS

“…A right R-module M is called weak CS if every semisimple submodule of M is essential in a summand of M . I. Amin, M. Yousif and N. Zeyada [1] introduced soc-injective and strongly soc-injective modules, for any two modules M and N , M is soc-N -injective if any R-homomorphism f : soc(N ) −→ M extends to N . R is called right (self-) soc-injective, if the right R-module R R is soc-injective.…”