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In this paper we show that if {Ri}i∈I is a family of commutative rings and {Mi}i∈I is a family of modules such that each Mi is an Ri-module, then the direct product Mi is a -module over Ri if and only if each Mi is an -module over Ri, i ∈ I. This result extends the work of Hong, Kim, Lee and Ryu in Rings with Property () and their extensions, J. Algebra315 (2007), 612–628. Moreover, we characterize Property () for modules in terms of Property () of their total quotient modules, extending the work of Dobbs and Shapiro in On the strong ()-ring of Mahdou and Hassani, Mediterr. J. Math.10 (2013), 1995–1997. [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]