×

Orders in rings without identity. (English) Zbl 0719.16022

The authors generalize the concept of a ring of quotients. An element a of a semigroup S is called left square cancellable if for any \(x,y\in S^ 1\), \(a^ 2x=a^ 2y\) implies \(ax=ay\). An element that is both left and right square cancellable is called square cancellable. For \(a\in S\), the group inverse of a, if it exists, is denoted by \(a^*\); it is the inverse of a in any subgroup of S containing a. A left order in a semigroup Q is a subsemigroup S such that every square cancellable element of S has a group inverse in Q and every element of Q is of the form \(a^*b\), for some a,b\(\in S\). The subsemigroup S is an order in Q if it is both a left and a right order in Q. In rings with identity, a left order in the new sense is a left order in the traditional sense. If the ring Q is (von Neumann) regular or left or right perfect, then a traditional order in Q is an order in Q in the new sense. The authors prove several results. Theorem 1: Let R be a subring of a ring Q with identity. If the multiplicative semigroup of R is a left order in the multiplicative semigroup of Q, then R is a traditional left order in Q. Theorem 2: A regular ring R is directly finite iff every nontrivial principal factor of R is completely 0-simple.

MSC:

16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
20M25 Semigroup rings, multiplicative semigroups of rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
20M10 General structure theory for semigroups
16U20 Ore rings, multiplicative sets, Ore localization
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Behrens E.A., Ring Theory (1972)
[2] Ben-Israel A., Generalized Inverses :Theory and Applications (1974)
[3] Clifford A.H., The Algebraic Theory of Semigroups, Mathematical Surveys 7 1 (1967) · Zbl 0178.01203
[4] DOI: 10.1007/BF01895723 · Zbl 0147.28602
[5] DOI: 10.1112/plms/s3-44.1.103 · Zbl 0481.20036
[6] Fountain J.B., Straight left orders in regular rings · Zbl 0774.16003
[7] Fountain J.B., Orders in semiprime rings with minimal condition for principal right ideals · Zbl 0774.16004
[8] Goodearl K.R., Von Neumann Regular Rings (1979)
[9] Howie J.M., An Introduction to Semigroup Theory (1976) · Zbl 0355.20056
[10] Jacobson N., Structure of Rings (1964)
[11] DOI: 10.1016/0021-8693(76)90158-7 · Zbl 0349.20025
[12] DOI: 10.1007/BF02194891 · Zbl 0299.20057
[13] Steinfeld O., Quasi-ideals in Rings and Semigroups (1978) · Zbl 0403.16001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.