Prüfer ⋆–multiplication domains and ⋆–coherence

Abstract

The purpose of this paper is to deepen the study of the Prüfer -multiplication domains, where is a semistar operation. For this reason, we introduce the -domains, as a natural extension of the v-domains. We investigate their close relation with the Prüfer -multiplication domains. In particular, we obtain a characterization of Prüfer -multiplication domains in terms of -domains satisfying a variety of coherent-like conditions. We extend to the semistar setting the notion of H-domain introduced by Glaz and Vasconcelos and we show, among the other results that, in the class of the H( )-domains, the Prüfer -multiplication domains coincide with the -domains.

Star operations on modules

“…This concept extends that of Prüfer * -multiplication domain (P * MD), for a given semistar operation * on D, [5]. By the above, a domain D is a (M, )-multiplication domain if and only if it is a ( f ) D -domain (i.e., a domain in which each finitely generated ideal is ( f ) D -invertible, see [11]). When =˜ we can say something more (cf.…”

Some remarks on Prüfer ⋆-multiplication domains and class groups

“…The equivalences (i)⇔(ii)⇔(iii) follow from [29,Proposition 11]. (iii)⇒(iv) Since a * -ideal is trivially also a * f -ideal, thus under the assumption (iii), Inv * (D) ⊆ Inv * f (D).…”

Descending chains of semistar operations

“…For example, Zafrullah ([23, Theorem 8]) proved that an integral domain R is a Prüfer domain if and only if R is an integrally closed domain such that the t-operation and d-operation coincide, and Wang and McCasland ( [16,Theorem 5.4]) proved that R is a Krull domain if and only if R is a completely integrally closed domain such that the w-operation and v-operation coincide. In a similar way, semistar operations have received significant interest in the field of multiplicative ideal theory (for instance, see [4], [6], [7], [8], [17], [19], [20], [21], to name only a few).…”