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The characteristic ring and the “best” way to adjoin a one

Part of: General commutative ring theory

Published online by Cambridge University Press:  09 April 2009

W. D. Burgess  and
P. N. Stewart
W. D. Burgess
Affiliation:
Department of Mathematics, University of OttawaOttawa, OntarioCanadaK1N 6N5
P. N. Stewart
Affiliation:
Department of Mathematics, Statistics and Computing Science, Dalhousie UniversityHalifax, Nova ScotiaCanadaB3H 3J5
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Abstract

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For any ring S we define and describe its characteristic ring, k(S). It plays the rôle of the usual characteristic even in rings whose additive structure, (S, +), is complicated. The ring k(S) is an invariant of (S, +) and also reflects certain non-additive properties of S. If R is a left faithful ring without identity element, we show how to use k(R) to embed R in a ring R1 with identity. This unital overring of R inherits many ring properties of R; for instance, if R is artinian, noetherian, semiprime Goldie, regular, biregular or a V-ring, so too is R1. In the case of regularity (or generalizations thereof), R1 satisfies a universal property with respect to the adjunction of an identity

MSC classification

Secondary: 13A99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 3 , December 1989 , pp. 483 - 496
Copyright
Copyright © Australian Mathematical Society 1989

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