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On valuation subalgebras and their centres

Published online by Cambridge University Press:  18 May 2009

C. P. L. Rhodes
C. P. L. Rhodes
Affiliation:
University of Wales College of Cardiff, Cardiff CF2 4AG, Wales
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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 1 , January 1989 , pp. 115 - 126
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

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