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Let R be a ring. A left R-module M (resp. right R-module N) is called singly injective (resp. singly flat) if Ext1( F / C, M) = 0 (resp. Tor1( N, F / C) = 0) for any cyclic submodule C of any finitely generated free left R-module F. It is shown that R is a left pseudo-coherentring if and only if any direct product of singly flat right R-modules is singly flat if and only if any direct limit of singly injective left R-modules is singly injective if and only if every right R-module has a singly flat preenvelope; and every left R-module over a left pseudo-coherent ring R has a singly injective cover. Some applications are also given. [ABSTRACT FROM AUTHOR]

In this paper, strongly Gorenstein flat modules are introduced and investigated. An R-module M is called strongly Gorenstein flat if there is an exact sequence … → P1 → P0 → P0 → P1 → … of projective R-modules with M D ker.P0 ! P1/ such that Hom.??; F/ leaves the sequence exact whenever F is a flat R-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied. [ABSTRACT FROM AUTHOR]

Let R be a ring. Recall that a left R-module M is coherent if every finitely generated submodule of M is finitely presented. R is a left coherent ring if the left R-module RR is coherent. In this paper, we say that R is left J-coherent if its Jacobson radical J(R) is a coherent left R-module. J-injective and J-flat modules are introduced to investigate J-coherent rings. Necessary and sufficient conditions for R to be left J-coherent are given. It is shown that there are many similarities between coherent and J-coherent rings. J-injective and J-flat dimensions are also studied. [ABSTRACT FROM AUTHOR]

In this paper, Gorenstein FP-injective modules are introduced and studied. An R-module M is called Gorenstein FP-injective if there is an exact sequence ⋯ → E1 → E0 → E0 → E1 → ⋯ of injective R-modules with M = ker(E0 → E1) and such that Hom(E, -) leaves the sequence exact whenever E is an FP-injective R-module. Some properties of Gorenstein FP-injective and Gorenstein flat modules over coherent rings are obtained. Several known results are extended. [ABSTRACT FROM AUTHOR]

Let R be a ring and M a right R -module. M is called n - FP -projective if Ext 1 ( M , N ) = 0 for any right R -module N of FP -injective dimension = n , where n is a nonnegative integer or n = 8. ? R ( M ) is defined as sup{ n : M is n - FP -projective} and ? R ( M ) = - 1 if Ext 1 ( M , N ) ? 0 for some FP -injective right R -module N . The right ?-dimension r .?-dim( R ) of R is defined to be the least nonnegative integer n such that ? R ( M ) = n implies ? R ( M ) = 8 for any right R -module M . If no such n exists, set r .?-dim( R ) = 8. The aim of this paper is to investigate n - FP -projective modules and the ?-dimension of rings. [ABSTRACT FROM AUTHOR]

This paper consists of two parts. In the first part, it is proven that a ring R is right PP if and only if every right R-module has a monic PI-cover, where PI denotes the class of all P-injective right R-modules. In the second part, for a nonempty subset X of a ring R, we introduce the notion of X-PP rings which unifies PP rings, PS rings and nonsingular rings. Special attention is paid to J-PP rings, where J is the Jacobson radical of R. It is shown that right J-PP rings lie strictly between right PP rings and right PS rings. Some new characterizations of (von Neumann) regular rings and semisimple Artinian rings are also given. [ABSTRACT FROM AUTHOR]

A ring R is called right AGP-injective if, for any 0 = a ∈ R, there exists n > 0 such that an = 0 and Ran is a direct summand of lr(an). In this paper some conditions which are sufficient or equivalent for a right AGP-injective ring to be von Neumann regular (right self-injective, semisimple) are provided. It is shown that a ring ft is von Neumann regular if and only if R is right AGP-injective and for any 0 = a ∈ R there exists a positive integer n with 0 = an such that anR is a projective right R-module if and only if R is a right AGP-injective ring whose divisible and torsionfree right R-modules are GP-injective. We also show that if R is a primitively finite right AGP-injective ring, then R ≅ R1 × R2, where R1 is semisimple and every simple right ideal of R2 is nilpotent. In addition, it is proven that if R is a right MI and right AGP-injective ring satisfying the a.c.c on right annihilators, then R is quasi-Frobenius. [ABSTRACT FROM AUTHOR]

Let R be a ring and g(x) a polynomial in C[x], where C=C(R) denotes the center of R. Camillo and Simón called the ring g(x)-clean if every element of R can be written as the sum of a unit and a root of g(x). In this paper, we prove that for a, b ∈ C, the ring R is clean and b-a is invertible in R if and only if R is g1(x)-clean, where g1(x)=(x-a)(x-b). This implies that in some sense the notion of g(x)-clean rings in the Nicholson–Zhou Theorem and in the Camillo–Simón Theorem is indeed equivalent to the notion of clean rings. [ABSTRACT FROM AUTHOR]