Research Article

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

Published in: Quaestiones Mathematicae
Volume 44 , issue 2 , 2021 , pages: 147–161

DOI: 10.2989/16073606.2019.1676838

Author(s): Samir Bouchiba , Morocco , Mouhssine El-Arabi , Morocco

Keywords: 13C13 , 13A99 , 13C13 , 13A99

Abstract

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. 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.

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