In this paper we show that a direct decomposition of modules M N, with N homologically independent to the inJective hull of H, is a CS-module if and only if N is injective relative to H and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for q...

In this paper, we investigate the notions of X ⊥ -projective, id="M2"> -injective, and id="M3"> -flat modules give some characterizations these modules, where id="M4"> is a class left modules. We prove that all id="M5"> -projective Kaplansky. Further, if id="M6"...

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let π be a group and let M → N be a homomorphism between projective Z[π]-modules such that Zp⊗Z[π]M → Zp⊗Z[π]N is injective; for which other right Z[π]-modules V is the induced map V ⊗Z[π] M → V ⊗Z[π] N also injective? Our main theorem gives a new criterion which combin...

Lexicographic constraints are commonly used to break variable symmetries. In the general case, the number of constraint to be posted is potentially exponential in the number of variables. For injective problems (AllDiff), Puget’s method[11] breaks all variable symmetries with a linear number of constraints. In this paper we assess the number of constraints for “almost” injective problems. We pr...

Let $R$ be a ring, $n$ an non-negative integer and $d$ positive or $\infty$. A right $R$-module $M$ is called \emph{$(n,d)^*$-projective} if ${\rm Ext}^1_R(M, C)=0$ for every $n$-copresented $C$ of injective dimension $\leq d$; ring \emph{right $(n,d)$-cocoherent} with $id(C)\leq d$ $(n+1)$-copresented; $(n,d)$-cosemihereditary} whenever $0\rightarrow C\rightarrow E\rightarrow A\rightarrow 0$ e...

For a left and right Noetherian ring $R$, we give some equivalent characterizations for $\_RR$ satisfying the Auslander condition in terms of flat (resp. injective) dimensions minimal injective coresolution resolution) $R$-modules. Furthermore, prove that an artin algebra $R$ condition, is Gorenstein if only subcategory consisting finitely generated modules contravariantly finite. As applicatio...

In this paper we show that a direct decomposition of modules M N, with N homologically independent to the inJective hull of H, is a CS-module if and only if N is injective relative to H and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for q...

We investigate preserving of projectivity and injectivity by the object-wise tensor product of RC-modules, where C is a small category. In particular, let O(G,X) be the category of canonical orbits of a discrete group G, over a G-set X. We show that projectivity of RO(G,X)-modules is preserved by this tensor product. Moreover, if G is a finite group, X a finite G-set and R is an integral domain...