Purity over Prüfer v-multiplication domains, II

Abstract

It is well known in [Absolutely pure modules, Proc. Amer. Math. Soc. 26 (1970) 561–566, Theorem 6] that a domain [Formula: see text] is a Prüfer domain if and only if every divisible [Formula: see text]-module is absolutely pure, where an [Formula: see text]-module [Formula: see text] is called absolutely pure if [Formula: see text] for every finitely presented [Formula: see text]-module [Formula: see text]. In this paper, we extend this result to Prüfer [Formula: see text]-multiplication domains (P[Formula: see text]MDs). To do this, comparing with [An Introduction to Homological Algebra, 2nd edn. (Springer, Science+Business Media, LLC, New York, 2009), Theorem 3.69], we firstly give homological characterizations of [Formula: see text]-purity, and we introduce the concept of absolutely [Formula: see text]-pure modules over commutative rings with zero divisors. Finally, we prove that a domain [Formula: see text] is a P[Formula: see text]MD if and only if every divisible [Formula: see text]-module is absolutely [Formula: see text]-pure, and compare absolutely [Formula: see text]-purity with absolutely purity by giving an example.