Tilted algebras, that is endomorphism algebras of tilting modules over a hereditary algebra, have been one of the main objects of study in representation theory of algebras since their introduction by Happel and Ringel [9]. As a generalization, Happel, Reiten and Smalø studied endomorphism algebras of tilting objects of a hereditary abelian category which they call quasi-tilted algebras [8]. We...

If for any maximal right ideal P of B and a ∈ N(B) ,aB/ aP is almost N-injective, then ring said to be generalized N-injective. In this article, we present some significant findings that are known N-injective rings demonstrate they hold rings. At the same time, study case in which every S.S.Right B-module

We will introduce the notion of Gorenstein category as the convenient setup for doing Gorenstein Homological Algebra in categories of sheaves, or in general in categories without enough projective objects. We will illustrate this notion by showing that the category of Qcoh(X) of quasi-coherent sheaves on a locally Gorenstein projective scheme fits into this setup. Then we will focus on the cate...

Minimality ranks in the style of Deissler are one way of measuring the structural complexity of minimal extensions of first-order structures. In particular, positive Deissler rank measures the complexity of the injective envelope of a module as an extension of that module. In this paper we solve a problem of the second author by showing that certain injective envelopes have the maximum possible...

We find the injective hulls of partially ordered monoids in the category whose objects are po-monoids and submultiplicative order-preserving functions. These injective hulls are with respect to a special class of monics called “embeddings”. We show as well that the injective objects with respect to these embeddings are precisely the quantales.

In this paper, motivated by Cagliari and Mantovani, we have obtained a characterization of injective objects (with respect to the class embeddings in category

we introduce the class of “right almost v-rings” which is properly between the classes of right v-rings and right good rings. a ring r is called a right almost v-ring if every simple r-module is almost injective. it is proved that r is a right almost v-ring if and only if for every r-module m, any complement of every simple submodule of m is a direct summand. moreover, r is a right almost v-rin...

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non–empty set is injective in classical ZF, which arg...

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...