It has been observed by different authors that QTAG-modules behave very much like torsion abelian groups. In this paper, in section 3, we characterize quasi-essential submodules (Theorem 3.9) and further find a characterization for an h-pure submodule to be a direct summand (Theorem 3.11). In section 4, we obtained a necessary and sufficient condition for a submodule to be contained in a minima...

Let S be the left bialgebroid End BAB over the centralizer R of a right D2 algebra extension A | B, which is to say that its tensor-square is isomorphic as A-B-bimodules to a direct summand of a finite direct sum of A with itself. Without an antipode, we prove that the left endomorphism algebra is a left S-Galois extension of A, and find a formula for the inverse Galois mapping. As a corollary,...

A module M is said to have the SIP if intersection of each pair direct summands also a summand M. In this article, we define SIPr and only exact where r left preradical for category right modules. We investigate structural properties SIPr-modules locate implications between other properties. deal with decomposition theory as well SIPr-modules. provide examples by looking at special preradicals.

We show that any quasi-Gorenstein deformation of a 3-dimensional Buchsbaum local ring with I-invariant 1 admits maximal Cohen-Macaulay module, provided it is quotient Gorenstein ring. Such class rings includes two instances unique factorization domains constructed by Marcel-Schenzel and Imtiaz-Schenzel, respectively. Apart from this result, motivated the small conjecture in prime characteristic...

A module $M$ is called a $C_{21}$-module if, whenever $A$ and $B$ are submodules of with $A \cong B$, nonsingular direct summand $M$, then $M$. Various examples $C_{21}$-modules presented. Some basic properties these modules investigated. It shown that the class rings $R$ over which every $C_2$-module exactly right SI-rings. Also, we prove for ring $R$, $R$-module has $(C_{21})$ if only t-semis...

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal or cosilting. The aim of this paper is twofold. Firstly, we determine the tilting cotilting over a tame hereditary algebra. In particular, show that large module if only it has an adic as direct summand. Secondly, discuss behavior minimality under ring extensions. We commutative noetherian extend to along any f...

In this note we show that the second homotopy group of B(2, G), classifying space for commutativity a compact Lie G, contains direct summand isomorphic to $$\pi _1(G)\oplus \pi _1([G,G])$$ , where [G, G] is commutator subgroup G. It follows from similar statement E(2, fiber canonical inclusion $$B(2,G)\hookrightarrow BG$$ . As consequence our main result obtain if G) 2-connected, then simply-co...

Our goal in this paper is to advance the state of art topic uniqueness unconditional basis. To that end we establish general conditions on a pair (X,Y) formed by quasi-Banach space X and Banach Y which guarantee every basis their direct sum X⊕Y splits into bases each summand. As application our methods obtain that, among others, spaces Hp(Td)⊕T(2) Hp(Td)⊕ℓ2, for p∈(0,1) d∈N, have unique (up equ...

In this paper, we describe $ss$-supplement submodules in terms of a special class endomorphisms. Let $R$ be ring with semisimple radical and $P$ projective $R-$module. We show that there is bijection between ss-supplement $End_{R}(P)$. Moreover, define radical-s-projective modules as generalization modules. prove every submodule $R-$module over the radical. $SSI$-ring $R$, projective. provide r...