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TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

Part of: Rings and algebras arising under various constructions Semigroups Rings and algebras with additional structure

Published online by Cambridge University Press:  21 July 2015

ZACHARY MESYAN  and
LIA VAŠ
ZACHARY MESYAN
Affiliation:
Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA e-mail: zmesyan@uccs.edu
LIA VAŠ
Affiliation:
Department of Mathematics, Physics and Statistics, University of the Sciences, Philadelphia, PA 19104, USA e-mail: l.vas@usciences.edu
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Abstract

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The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

MSC classification

Primary: 16S36: Ordinary and skew polynomial rings and semigroup rings
Secondary: 16W10: Rings with involution; Lie, Jordan and other nonassociative structures 16S50: Endomorphism rings; matrix rings 20M25: Semigroup rings, multiplicative semigroups of rings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016 , pp. 97 - 118
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

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