On valuation subalgebras and their centres

Abstract

We shall extend results of Samuel [19] and Griffin [8, 9] about conditions which generalise the notion of valuation domain in a field. Let U be a commutative ring with identity, R a subring of U and L an R-submodule of U. The conditions we study have in common the property (EV), that the submodules L:x (x ∈ U) form a chain. We pay particular attention to the strongest of the conditions, viz, that L be a Manis valuation (MV) subring, i.e. having a prime ideal P such that (L, P) is a maximal pair in U (see [19], [16] and e.g. [4]). Such P is unique, being the union of all L:x such that x ∉ L, which we call P+(L) the centre of L. This set P+ plays a key role in the study of all our valuation conditions.