A right $R$-module $M$ is called $R$-projective provided that it is projective relative to the right $R$-module $R_R$. This paper deals with the rings whose all nonsingular right modules are $R$-projective. For a right nonsingular ring $R$, we prove that $R_R$ is of finite Goldie rank and all nonsingular right $R$-modules are $R$-projective if and only if $R$ is right finitely $\Sigma$-$CS$ and flat right $R$-modules are $R$-projective. Then, $R$-projectivity of the class of nonsingular injective right modules is also considered. Over right...

In this note we obtain a necessary and sufficient condition for a ring to be $s$-weakly regular (i) When $R$ is a ring with identity and without divisors of zero (ii) When $R$ is a ring without divisors of zero. Further it is proved in a $s$-weakly regular ring with identity and without units every element is a zero divisor.

Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x ∈ M there exists a homomorphism F: M → R such that f(x) = x. Let Q be a left R-module and h: Q → M a homomorphism. We call h locally split if for every x ∈ M there exists a homomorphism g: M → Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:(1) M is Zelmanowitz-regular.(2) every homomorphism into M...

Let $n,d$ be two non-negative integers. A left $R$-module $M$ is called $(n,d)$-injective, if ${\mathrm{Ext}}^{d+1}(N,M)=0$ for every $n$-presented left $R$-module $N$. A right $R$-module $V$ is called $(n,d)$-flat, if ${\mathrm{Tor}}_{d+1}(V,N)=0$ for every $n$-presented left $R$-module $N$. A left $R$-module $M$ is called weakly $n$-$FP$-injective, if ${\mathrm{Ext}}^n(N,M)=0$ for every $(n+1)$-presented left $R$-module $N$. A right $R$-module $V$ is called weakly $n$-flat, if ${\mathrm{Tor}}_n(V,N)=0$ for every $(n+1)$-presented left $R$-module $N$. In this paper, we give some characterizations and properties of $(n,d)$-injective modules and $(n,d)$-flat modules in the cases...

Let $𝒲$ be a self-orthogonal class of left $R$-modules. We introduce a class of modules, which is called strongly $𝒲$-Gorenstein modules, and give some equivalent characterizations of them. Many important classes of modules are included in these modules. It is proved that the class of strongly $𝒲$-Gorenstein modules is closed under finite direct sums. We also give some sufficient conditions under which the property of strongly $𝒲$-Gorenstein module can be inherited by its submodules and quotient modules....

An associated ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products. We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is FTF if and only if Re is left FTF, where e is a neutral element of G. This provides new examples of left FTF rings. Some consequences of this Theorem are given.

Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(𝒯,n)$-injective if ${\mathrm{Ext}}_R^n(C,M)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(𝒯,n)$-flat if ${\mathrm{Tor}}_n^R(M,C)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(𝒯,n)$-projective if ${\mathrm{Ext}}_R^n(M,N)=0$ for each $(𝒯,n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(𝒯,n)$-coherent if whenever $0\to K\to P\to C\to 0$ is exact, where $C$ is $(𝒯,n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(𝒯,n)$-projective; the ring $R$ is called $(𝒯,n)$-semihereditary...

A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K. C. O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective modules over a simple regular ring.

We introduce the abelian category R-gr of groupoid graded modules and give an answer to the following general question: If U: R-gr → R-mod denotes the functor which associates to any graded left R-module M the underlying ungraded structure U(M), when does either of the following two implications hold: (I) M has property X ⇒ U(M) has property X; (II) U(M) has property X ⇒ M has property X? We treat the cases when X is one of the properties: direct summand, free, finitely generated, finitely presented,...