… LA M-simple-injective if for any submodule N of … perfect ring A, we will show that eA^ is
A-simple-injective if and only if there exists a local idempotent / in A such that (eA, Af) is an i-pair …

… , and if e, f ∈ E, we say that the pair (eR, Rf) is an i-pair if soc(eR) ∼= fR/fJ and soc(Rf) ∼=
Re/Je. … Perfect ring, self-injective ring, quasi-Frobenius ring. The research of both authors was …

… module L over a left or right perfect ring A to be injective. Also, by developing the … ] to left
perfect rings and provide simple proofs of them. Throughout this note, rings are associative rings …

… ring every component is the injective envelop of its minimal ideal we obtain: … Our next aim
is to prove the existence of Artinian rings R of left injective dimension 1, evenso 1. inj. dim., R/…

… and a right IP-injective ring is not necessarily left IP-injective, … if R is left perfect, left GIN and
right simple-injective then R is … a ring R a left GIN-ring if r(I ∩ K) = r(I) + r(K) for each pair of left …

… When left self-injective rings are studied our aim is to derive chain conditions of these under
… that βιΑβ] c: J(A) for all pairs i Φ /. The results below are only proved for reduced artinian …

… left self injective, semi-perfect ring is a left q-ring. The following example shows this is not
so. … Then R has the property that for each pair e, f of orthooonal idempotents eR f ~_ Soc(~R). …

… (These results imply that not every valuation ring (VR) is quotient-injective, since otherwise
every factor ring of a VR would be quotient-injective, hence FSI whence almost maximal.) …

… 8 and 9 we review some results on IF rings and Σ-injective rings, respectively. … if and only
if there exists a primitive idempotent f ∈ R such that (eR, Rf) is an injective pair. In this case, …

… rings R possess the property that their weak-injective modules … of which is the injective or
the weak-injective envelope of a … pairs involving the classes of weak-injective and injective …