article · 2015
SEMIHEREDITARY MONOID RINGS
James Kuzmanovich, Mark L. Teply, James Kuzmanovich, Mark L. Teply
Cited by 0Abstract and details only
Abstract
Hereditary and semihereditary rings have been the subject of considerable study. Many interesting examples of these rings arise as group rings or semigroup rings. Recently Chouinard, Hardy, and Shores [3] have completely determined the commutative semihereditary monoid rings. Goursaud and Valette [8] have given results concerning hereditary and semihereditary group rings, while Dicks [5] has completely determined the hereditary group rings. In this paper we continue the study of semihereditary monoid rings. Let R be a not necessarily commutative ring with identity, and let S be a monoid. We characterize in terms of R and S the left semihereditary monoid rings R[S] when S is a semilattice of torsion groups. This yields a characterization of the left semihereditary monoid rings R[S] when S is commutative and periodic. If S is a non-periodic commutative monoid, we show that a necessary condition for R[S] to be left semihereditary is that Z(R)[S] be semihereditary and R be yon Neumann regular. In view of results of [3], this yields much information about the structure of S. If, in addition, S is a strong semilattice of commutative groups, then necessary and sufficient conditions for R[S] to be left semihereditary are obtained. Our main tool is a technical lemma (Lemma 3), which may have independent interest in determining whether or not a ring is semihereditary. Throughout this paper all rings will be associative with identity. We will use Z(R) to denote the center of a ring R, and œR(X) to denote the left annihilator in R of an element x. The ring of n X n matrices over R will be denoted by Mn(R). A semilattice Y is a commutative semigroup all of whose elements are idempotent; such a Y has a natural ordering denoted by •<. In [10], [17] and other papers, a ring R is called a supplementary semilattice sum of rings R a (a C Y) if the following conditions hold: R = Zo•cyRo•, Ro•R$3 _C Ro•s3 for all o•,S3 • Y, and Ro • C•
Cite this paper
Kuzmanovich, J., Teply, M. L., Kuzmanovich, J., & Teply, M. L. (2015). SEMIHEREDITARY MONOID RINGS.
Kuzmanovich, James, et al. “SEMIHEREDITARY MONOID RINGS.” 2015.
Kuzmanovich, James, Mark L. Teply, James Kuzmanovich, and Mark L. Teply. 2015. “SEMIHEREDITARY MONOID RINGS.”
Kuzmanovich, J. et al. (2015) ‘SEMIHEREDITARY MONOID RINGS’.
@article{kuzmanovich2015semihereditary,
title = {{SEMIHEREDITARY MONOID RINGS}},
author = {Kuzmanovich, James and Teply, Mark L. and Kuzmanovich, James and Teply, Mark L.},
year = {2015}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.