## 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.)
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### References:

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