On property (𝒜) for modules over direct products of rings

Abstract

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. Algebra 315 (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.