On v-domains: a survey

Abstract

The v-domains generalize Prüfer and Krull domains and have appeared in the literature with different names. This paper is the result of an effort to put together information on this useful class of integral domains. In this survey, we present old, recent and new characterizations of v-domains along with some historical remarks. We also discuss the relationship of v-domains with their various specializations and generalizations, giving suitable examples.

Topological Aspects of Irredundant Intersections of Ideals and Valuation Rings

“…An integral domain A is a v-domain if whenever I, J, K are ideals of A such that (IK) v = (JK) v , then I v = J v . Examples of such domains include completely integrally closed domains and Prüfer vmultiplication domains; for a recent survey of this class of rings, see [17]. A vdomain A has a unique maximal Kronecker function ring [22,Theorem 28.1], so by (4.14), A has a unique minimal representation.…”

Anneaux à diviseurs et anneaux de Krull (une approche constructive)

“…, v j,n A contient une puissance c mj j Bilan : d'une part lorsqu'on inverse un v j,i l'idéal devientégalà a i , et d'autre part l'idéal engendré par les v j,i contient les c mj j , donc il est de profondeur 2. Le noyau H de la projection canonique d'un groupe réticulé G sur un groupe réticulé quotient G/H est ce que l'on appelle un sous-groupe solide 11 , c'est-à-dire un sous-groupe vérifiant la propriété : ξ ∈ H et |ζ| |ξ| impliquent ζ ∈ H.…”

Star operation on orders in simple Artinian rings

“…In turn, star operations are a powerful tool used to study multiplicative ideal theory. Most progress, however, is concerned with the application of star operations in the commutative setting, see [2,11,12,27] and references therein, and relatively little is known in the non-commutative case. To the best of our knowledge, the first to advance the latter were Asano and Murada [4], who, in 1953, used the ν-operation to study the Arithmetic properties of noncommutative semigroups.…”