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**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.

Reviewer: A.A.Iskander (Lafayette)

### 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.) |

### Keywords:

ring of quotients; left square cancellable; right square cancellable; left order; semigroup; square cancellable element; multiplicative semigroup; regular ring; completely 0-simple
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\textit{J. Fountain} and \textit{V. Gould}, Commun. Algebra 18, No. 9, 3085--3110 (1990; Zbl 0719.16022)

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