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The characteristic ring and the “best” way to adjoin a one. (English) Zbl 0692.16019

For any ring S with unit element, the characteristic ring \(\kappa\) (S) is defined to be the maximal epic extension of the canonical image of \({\mathbb{Z}}\) in S. The structure of the epimorphs of \({\mathbb{Z}}\) is known, and it turns out, that \(\kappa\) (S) depends only on the additive structure of S. This makes it possible to define \(\kappa\) (R) for rings without unit as well. The authors use this fact to introduce a new way of adjoining a unit element to a ring: If R is left faithful, then R embeds in \(End(R_ R)\) and \(R^ 1\) is the subring \(R+\kappa (End(R_ R))\) of \(End(R_ R)\). \(R^ 1\) is in a certain sense not too big; e.g. R is dense as a right ideal of \(R^ 1\). Furthermore, \(R^ 1\) is a factor ring of the ring \(R^*=R\times \kappa (R)\), which, with the usual multiplication, is another unitary overring of R. The ring \(R^ 1\) (and sometimes also the ring \(R^*)\) inherits many properties of R. If a (left faithful) ring R is prime, semiprime, semiprime Goldie, noetherian, artinian, regular, strongly regular, fully idempotent or biregular, then so is \(R^ 1\). The same holds for \(\pi\)-regular rings, provided R is embeddable in a unitary \(\pi\)-regular ring. Finally, it is shown that in the context of generalized regularity, the ring \(R^ 1\) satisfies a universal property.
Reviewer: H.H.Storrer

MSC:

16S20 Centralizing and normalizing extensions
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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