J Austral Math Soc Ser A 47 pp483--496, 1989.

The Characteristic Ring and the "Best" Way to Adjoin a One

W. D. Burgess and P. N. Stewart

Abstract

For any ring S we define and describe its characteristic ring, k(S). It plays the role 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 os R¹. In the case of regularity (or generalizations thereof), R¹ satisfies a universal property with respect to the adjunction of an identity.

1980 AMS Subject Classification (1985 Revision): 16A56, 16A30, 13A99

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Authors

W. D. Burgess

Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5.

P. N. Stewart

Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5.