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Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45-67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules. [ABSTRACT FROM AUTHOR]

We present an answer to a problem raised by Anderson and Chun in [D. D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16(7) (2017) 1750143] on characterizing when the idealization R ⋉ M of a ring R on an R -module M is an 𝒜 -ring (respectively, an 𝒮 𝒜 -ring) in terms of module-theoretic properties of R and M. Also, we are concerned with an open question asked by these two authors which reads the following: What modules over a given ring R are homomorphic images of modules satisfying the strong Property 𝒜 ? (see, Question 4.4(1) in the above mentioned paper). This paper highly contributes to answer such a question. [ABSTRACT FROM AUTHOR]