Regularity of semigroup rings

Sufficient conditions are obtained in order that the semigroup rings of some semigroups be regular. For some inverse, orthodox and completely simple semigroups, the necessary and sufficient conditions for the regularity of the semigroup rings are found. The author wishes to thank Prof. Y. Q. Guo for his best help and advice. Key Words and Phrases: (von Neumann) regularity, semigroup ring, inverse semigroup, orthodox semigroup, completely [0-] simple semigroup.     

Duality for semigroup rings

Given a square n-matrix F and an n-row matrix G, pole-shifting problems consist in obtaining more or less arbitrary characteristic polynomials for F+GK, for suitable (“feedback”) matrices K. A review of known facts is given, various partial results are proved, and the case n = 2 is studied in some detail.     

On regular semigroup rings

Communicated by L. N. Shevrin     

The regularity of munn rings and semigroup rings

In this paper, the author studies the regularity of Munn rings and completely 0-simple semigroup rings. When either (i)R is a ring with identity andI and are infinite, or (ii)R is a strong IBN and fully Dedekind-finite ring with identity and eitherI or is finite, in Section 2, the regularity of the Munn ringM(R; I, ; P) is characterized; in Section 3, for a completely 0-simple semigroupS =M°(G; I, ; P), the regularity ofRS is characterized. Meantime, the author shows that for a locally finite monoidS and a ringR with identity,RS is a strong IBN ring if and only ifR is so;RS is fully Dedekind-finite if and only ifR is so.     

A note on the units of semigroup rings

In [2] Coleman and Enochs obtained results about the units of the polynomial ring R[x] for rings R satisfying a condi-tion which is, in some sense, a generalization of commutativity. In [3] some of these results were extended to group rings over an ordered group. In this note a class of rings larger than the class considered in [2] is used to extend the results in [2] and 3] to the semigroup ring RG, G an u.p, semigroup.

In the last section we give a necessary and sufficient condi-tion for an element to be a divisor of zero in RG where G is an u.p. semigroup.     

Identities of orthodox semigroup rings

In this paper, the conditions under which an orthodox semigroup ring possesse an identity are investigated. The author is grateful to the referee for his careful modification of this paper and helpful comments which resulted in several improvements.     

Arithmetical and semihereditary semigroup rings

In this paper conditions on the commutative ring R (with identity) and the commutative semigroup ring S (with identity) are found which characterize those semigroup rings R[S] which are reduced or have weak global dimension at most one. Likewise, those semigroup rings R[S] which are semihereditary are completely determined in terms of R and S.     

Radicals of semigroup rings of orthogroups

Communicated by L. N. Shevrin