Let R be a ring and let M be a right R-module with S End MR . M is called almost general quasiprincipally injective or AGQP-injective for short if, for any 0/ s ∈ S, there exist a positive integer n and a left ideal Xsn of S such that s / 0 and lS Ker s Ss ⊕ Xsn . Some characterizations and properties of AGQP-injective modules are given, and some properties of AGQP-injective modules with additi...

Let Λ be a left and right noetherian ring and mod Λ the category of finitely generated left Λ-modules. In this paper we show the following statements: (1) For a positive integer k the condition that the subcategory of mod Λ consisting of i-torsionfree modules coincides with the subcategory of mod Λ consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. (2) If Λ is an Auslande...

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 model theory of abelian groups was developed by Szmielew ([28] quantifier elimination and decidability) and Eklof & Fisher [4], who observed that K1saturated abelian groups are pure injective. Eklof & Fisher related the structure theory of pure injective abelian groups with their model theory. The extension of this theory to modules over arbitrary rings became possible after the work of Bau...

In this paper, we study the relation between m-strongly Gorenstein projective (resp. injective) modules and n-strongly Gorenstein projective (resp. injective) modules whenever m 6= n, and the homological behavior of n-strongly Gorenstein projective (resp. injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Goren...

In act theory, Pseudo injective acts and their generalizations are essential. As a result, the purpose of this work is to give generalization pseudo quasi principally specifically small acts. Besides, concept was presented, which can be employed later. If each S-monomorphism from M-cyclic sub-act M\textsubscript{S} N\textsubscript{S} extended S-homomorphism N\textsubscript{S}, An S-act called p...

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of RR into R (N) R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is...

We obtain some methods to construct a (strongly) proper resolution (resp. coproper coresolution) of one end term in a short exact sequence from that of the other two terms. By using this method, we prove that for a left and right Noetherian ring R, RR satisfies the Auslander condition if and only if so does every flat left R-module, if and only if the injective dimension of the ith term in a mi...

A right R-module M is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [25], Theorem 3. In particular, it is shown that a module M is g.q.f.d. iff every direct sum of M -singular M -inject...

Abstract Assume that k is an algebraically closed field and A a finite-dimensional wild -algebra. Recently, L. Gregory M. Prest proved in this case the width of lattice all pointed -modules undefined. Hence result Ziegler implies there exists super-decomposable pure-injective -module, if base countable. Here we give different straightforward proof fact. Namely, show special family -modules, cal...