Radicals of semigroup rings of commutative semigroups

Communicated by J. S. Ponizovskii     

Generators and relations of abelian semigroups and semigroup rings

The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show that the number of relations defining S is greater than or equal to the least number of generators of S minus the rank of the associated group of S. If equality holds, we say S is a complete intersection. The main part of this study is devoted to semigroups of natural numbers generated by 3 elements. These semigroups are complete intersections if and only if they are symmetric in the sense of R. Apéry [1]. This result applies to algebraic geometry: An affine space-curve C with the parametric equations x=t^a, y=t^b, z=t^c, a, b, c natural numbers with greatest common divisor 1, is a global idealtheoretic complete intersection, if and only if the semigroup S generated by a, b, c is symmetric.This paper forms part of the author's thesis, submitted at Lousiana State University.The writing of this paper was partially supported by NSF grant GP-6388 in which the author participated as a junior assistant at Purdue University.     

Flatness and the Ore condition for rings

We prove the following result on the universal localization of a ring at an ideal : If the universal localization is flat as an -module, then satisfies the Ore condition with respect to the multiplicative set of elements that become invertible modulo . It is well known that for domains the converse of this result holds, and hence we have found in this case a new characterization of the Ore condition.

     

Amenability and uniqueness for groupoids associated with inverse semigroups

We investigate recent uniqueness theorems for reduced \(C^*\)-algebras of Hausdorff étale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full \(C^*\)-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.     

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.     

Identities of semigroup rings over semilattices of completely (0-) simple semigroups

LetR be a ring with identity,S be a semigroup with the set of idempotentsE(S), and denote (E(S)) for the subsemigroup ofS generated byE(S). In this paper, we prove that ifS is a semilattice of completely 0-simple semigroups and completely simple semigroups, then the semigroup ringRS possesses an identity iff so doesR(E(S)); especially, the result is true forS being a completely regular semigroup.     

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.     

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.     

Amenability of semigroups and their algebras modulo a group congruence

We investigate the amenability of the semigroup algebras \({\ell^1(S/\rho)}\) , where \({\rho}\) is a group congruence (not necessarily minimal) on a semigroup S. We relate this to a new notion of amenability of Banach algebras modulo an ideal, to prove a version of Johnson’s theorem for a large class of semigroups, including inverse semigroups, E-inversive semigroup and E-inversive E-semigroups.     

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.     

On the derivations of semigroup rings

In questo lavoro studiamo i generatori del modulo delle derivazioni di anelli di semigruppoR=k[S], dandone una caratterizzazione sotto opportune ipotesi suS.     

Equivalent semigroup properties for the curvature-dimension condition

Some equivalent gradient and Harnack inequalities of a diffusion semigroup are presented for the curvature-dimension condition of the associated generator. As applications, the first eigenvalue, the log-Harnack inequality, the heat kernel estimates, and the HWI inequality are derived by using the curvature-dimension condition. The transportation inequality for diffusion semigroups is also investigated.     

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.     

On the weak regularity of semigroup rings

Sufficient conditions are obtained under which the semigroup ring of a semigroup, in particular, of an inverse semigroup, is weakly regular. For some inverse semigroups and some orthodox semigroups, the necessary and sufficient conditions for the weak regularity of the semigroup rings are found. A problem is mentioned especially which is much more difficult for weak regularity than for regularity. Only a partial solution of this problem is given for weak regularity.