an r-module m is called strongly noncosingular if it has no nonzero rad-small (cosingular) homomorphic image in the sense of harada. it is proven that (1) an r-module m is strongly noncosingular if and only if m is coatomic and noncosingular; (2) a right perfect ring r is artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

LetR be a ring andM a rightR-module with S= End(MR). The moduleM is called almost principally quasi-injective (or APQ-injective for short) if, for any m∈M, there exists an S-submodule Xm of M such that lMrR(m) = Sm ⊕ Xm. The module M is called almost quasiprincipally injective (or AQP-injective for short) if, for any s∈ S, there exists a left ideal Xs of S such that lS(ker(s)) = Ss ⊕ Xs. In thi...

The notions of purity and equational compactness of universal algebras have been studied by Banaschewski and Nelson. Also, Banaschewski deals with these notions in the special case of G-sets for a group G. In this paper we study these and related concepts in the category PRO of projection algebras, that is in N∞-sets, for the monoid N∞ with the binary operation m.n = min{m, n}. We show that eve...

Abstract Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ . $\Lambda an -order such that $Q\Lambda is separable -algebra. Maranda showed there exists $k\in \mathbb {N}$ for all -lattices L M , if $L/L\pi ^k\simeq M/M\pi ^k$ then $L\simeq M$ Moreover, complete indecomposable -lattice, also indecomposable. We extend Maranda’s theorem to the clas...

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...

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

We give a criterion for the existence of a super-decomposable pure-injective module over an arbitrary serial ring.

Let be a left and right noetherian ring and mod the category of finitely generated left -modules. In this article, we show the following results. 1 For a positive integer k, the condition that the subcategory of mod consisting of i-torsionfree modules coincides with the subcategory of mod consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. 2 If is an -Gorenstein ring and N...