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Turkish Journal of Mathematics

YUSUF ALAGÖZ

DOI

10.55730/1300-0098.3269

Abstract

Given modules $A_{R}$ and $_{R}B$, $_{R}B$ is called absolutely $A_{R}$-pure if for every extension $_{R}C$ of $_{R}B$, $A\otimes B\rightarrow A\otimes C$ is a monomorphism. The class $\underline{\mathfrak{Fl}}^{-1}(A_{R})=$\{$_{R}B$ : $_{R}B$ is absolutely $A_{R}$-pure\} is called the absolutely pure domain of a module $A_{R}$. If $_{R}B$ is divisible, then all short exact sequences starting with $B$ is RD-pure, whence $B$ is absolutey $A$-pure for every $RD$-flat module $A_{R}$. Thus the class of divisible modules is the smallest possible absolutely pure domain of an $RD$-flat module. In this paper, we consider $RD$-flat modules whose absolutely pure domains contain only divisible modules, and we referred to these RD-flat modules as $rd$-indigent. Properties of absolutely pure domains of $RD$-flat modules and of $rd$-indigent modules are studied. We prove that every ring has an $rd$-indigent module, and characterize $rd$-indigent abelian groups. Furthermore, over (commutative) SRDP rings, we give some characterizations of the rings whose nonprojective simple modules are $rd$-indigent.

Keywords

RD-flat modules, absolutely pure domains, $rd$-indigent modules, QF-rings

First Page

2292

Last Page

2303

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