Principally Small Injective Rings

Abstract

A right ideal I of a ring R is small in case for every proper right ideal K of R, K + I = R. A right R-module M is called P S-injective if every R-homomorphism f : aR → M for every principally small right ideal aR can be extended to R → M . A ring R is called right P S-injective if R is P S-injective as a right R-module. We develop, in this article, P S-injectivity as a generalization of P -injectivity and small injectivity. Many characterizations of right P S-injective rings are studied. In light of these facts, we get several new properties of a right GP F ring and a semiprimitive ring in terms of right P S-injectivity. Related examples are given as well.

SAS-Injective Modules

“…(1) All small-injective modules are SAS-injective, but the converse is not true in general, for example: let 𝑅 be the localization ring of ℤ at the prime 𝑝, that is 𝑅 = ℤ (𝑝) = { 𝑚 𝑛 : 𝑝 does not divide n}. Then 𝑅 is not small injective with soc(𝑅 𝑅 ) = 0 (see [13,Example 4]). Since soc(𝑅 𝑅 ) = 0, we have that Sa(𝑅 𝑅 ) = 0 and hence the zero ideal is the only semiartinian small right ideal in 𝑅 𝑅 .…”

SS-Injective Modules and Rings

“…Recall that a right -module is called mininjective [2] ( resp. small injective [3], principally small injective [4]) if every -homomorphism from any simple (resp. small, principally small) right ideal to extends to .…”

Almost Principally Small Injective Rings

“…principally, small, minimal) right ideal I, can be extended to R. The detailed discussion of Pinjective, small injective and mininjective rings can be found in [2,3,4,8,9,10,12]. The concept of P S-injective rings was first introduced in [14] as a generalization of P -injective rings and small injective rings. It was shown that every right P S-injective ring is also right mininjective.…”