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

In this paper we show that if {R_i}_i∈I is a family of commutative rings and {M_i}_i∈I is a family of modules such that each M_i is an R_i-module, then the direct product M_i is a -module over R_i if and only if each M_i is an -module over R_i, 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]