Let S be a semigroup generated by periodic elements and k be an infinite field. Suppose that the semigroup algebra kS has 1. The authors prove interesting properties of kS and S assuming that U(kS) satisfies a group identity (kS is called a GI-ring for short). In particular, S is locally finite and kS satisfies a polynomial identity. Thus Hartley's conjecture is confirmed for such semigroup algebras. If, furthermore, S is generated by a finite number of periodic elements, then they give necessary and sufficient conditions on S for kS to be a GI-ring. The authors obtain this result by proving first that if A is a semiprime GI-algebra over an infinite commutative domain D, such that A is generated as a D-algebra by U(A) and no element of D is a zero divisor in A, then A is a reduced ring. The latter fact is also used to give shorter proofs of some known results on group rings which are GI-rings.
Dooms, A, Jespers, E & Juriaans, SO 2005, 'On Group Identities for the Unit Group of Algebras and Semigroup Algebras over an Infinite Field', Journal of Algebra, vol. 284, pp. 273-283. <http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WH2-4DBCBRY-4&_user=1011600&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050280&_version=1&_urlVersion=0&_userid=1011600&md5=2ec26777e8cc0e9c5c0b27a84b7c0de8>
Dooms, A., Jespers, E., & Juriaans, S. O. (2005). On Group Identities for the Unit Group of Algebras and Semigroup Algebras over an Infinite Field. Journal of Algebra, 284, 273-283. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WH2-4DBCBRY-4&_user=1011600&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050280&_version=1&_urlVersion=0&_userid=1011600&md5=2ec26777e8cc0e9c5c0b27a84b7c0de8
@article{1a5099e1751b4226b3d38ec27a2ccc4c,
title = "On Group Identities for the Unit Group of Algebras and Semigroup Algebras over an Infinite Field",
abstract = "Let S be a semigroup generated by periodic elements and k be an infinite field. Suppose that the semigroup algebra kS has 1. The authors prove interesting properties of kS and S assuming that U(kS) satisfies a group identity (kS is called a GI-ring for short). In particular, S is locally finite and kS satisfies a polynomial identity. Thus Hartley's conjecture is confirmed for such semigroup algebras. If, furthermore, S is generated by a finite number of periodic elements, then they give necessary and sufficient conditions on S for kS to be a GI-ring. The authors obtain this result by proving first that if A is a semiprime GI-algebra over an infinite commutative domain D, such that A is generated as a D-algebra by U(A) and no element of D is a zero divisor in A, then A is a reduced ring. The latter fact is also used to give shorter proofs of some known results on group rings which are GI-rings.",
author = "Ann Dooms and Eric Jespers and Juriaans, {Stanley Orlando}",
note = "Journal of Algebra 284 no. 1 (2005), 273--283.",
year = "2005",
language = "English",
volume = "284",
pages = "273--283",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "Academic Press Inc.",
}