Mininjective Rings

Abstract

A ring R is called right mininjective if every isomorphism between simple right ideals is given by left multiplication by an element of R. These rings are shown to be Morita invariant. If R is commutative it is shown that R is mininjective if and only if it has a squarefree socle, and that every image of R is mininjective if and only if R has a distributive lattice of ideals. If R is a semiperfect, right mininjective ring in which eR has nonzero right socle for each primitive idempotent e, it is shown that R admits a Nakayama permutation of its basic idempotents, and that its two socles are equal if every simple left ideal is an annihilator. This extends well known results on pseudo-and quasi-Frobenius rings.

Semiperfect min-CS rings

“…Then it follows from [13,Proposition 3.3] that if e is a local idempotent, then Re has simple socle. Let now R Re 1 È F F F È Re n , where the e i are local idempotents and let C be the socle of Re j .…”

“…Conversely, if oeR T 0 for every local idempotent e of R, then we may apply [13,Theorem 3.7] to deduce that R is a left Kasch ring. Furthermore, (a) and (b) also follow from [13,Theorem 3.7].…”

“…Conversely, assume that (i) holds. Then R is a mininjective ring by [13,Proposition 4.12], and so the dual of every simple R-module is simple. Then it follows from Theorem 2.1 that R is semiperfect with essential socle, and R satis®es the C2-condition by [17,Lemma 1.15].…”

“…Instead of left CS rings we consider the much larger class of left min-CS rings (cf. [13]), i.e., rings R such that every minimal left ideal is essential in a direct summand and we show that this weak injectivity property is useful to obtain semiperfect rings. Indeed, we prove in Theorem 2.1 that if R is left min-CS, then the dual of every simple right R-module is simple if and only if R is semiperfect with o R R oR R and oRe is simple and essential for every local idempotent e of R. Thus we establish the following pattern: we work with an injectivity condition on the left and a``cogenerating'' condition on the right, both closely related to simple modules, and we try to prove that R is semiperfect and, in some cases, that R is in a certain sense close to being left PF.…”

“…R is said to be a right Kasch ring when every simple right R-module embeds in R R . R is called right mininjective [13] if every R-homomorphism from a minimal right ideal of R into R is given by left multiplication by an element of R. By [13, Proposition 2.2] , the dual of every simple right R-module is simple if and only if R is a right Kasch right mininjective ring.…”