Abstract
In 1976, Domanov showed that the algebra of an inverse semigroup S over a field F is semiprimitive (that is, has zero Jacobson radical) if the algebra of each maximal subgroup of S over F is semiprimitive. It is known that the converse statement is false in general. The principal purpose of this paper is to announce that if the semi-lattice of S satisfies a certain finiteness condition, introduced by Teply, Turman and Quesada in 1980, then the converse does hold. Corresponding results for primitivity are also discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S.A. Amitsur. ‘On the semi-simplicity of group algebras.’ Michigan Math. J. 6 (1959), 251–253.
A.H. Clifford and G.B. Preston. The algebraic theory of semigroups. Math. Surveys 7 (American Math. Soc, Providence R.I., 1961 (Vol. I) and 1967 (Vol. II)).
O.I. Domanov. ‘On the semisimplicity and identities of inverse semigroup algebras.’ Rings and modules. Matem. Issled., Vyp. 38 (1976), 123–137. [In Russian].
E. Formanek. ‘Group rings of free products are primitive.’ J. Algebra 26 (1973), 508–511.
W.D. Munn. ‘On semigroup algebras.’ Proc. Cambridge Philos. Soc. 51 (1955), 1–15.
W.D. Munn. ‘Matrix representations of semigroups.’ Proc. Cambridge Philos. Soc. 53 (1957), 5–12.
W.D. Munn. ‘Free inverse semigroups.’ Proc. London Math. Soc (3) 29 (1974), 385–404.
W.D. Munn. ‘Semiprimitivity of inverse semigroup algebras.’ Proc. Roy. Soc. Edinburgh A 93 (1982), 83–98.
W.D. Munn. ‘Nil ideals in inverse semigroup algebras.’ Submitted to J. London Math. Soc.
W.D. Munn. ‘Two examples of inverse semigroup algebras.’ Submitted to Semigroup Forum.
V.A. Oganesyan. ‘On the semisimplicity of a system algebra.’ Akad. Nauk Armyan. SSR Dokl. 21 (1955), 145–147. [In Russian]
D.S. Passman. ‘On the semisimplicity of twisted group algebras.’ Proc. Amer. Math. Soc. 25 (1970), 161–166.
D.S. Passman. The algebraic structure of group rings (Wiley — Interscience, New York, 1977).
I.S. Ponizovskiĭ. ‘On matrix representations of associative systems.’ Mat. Sbornik 38 (1956), 241–260. [In Russian].
I.S. Ponizovskiĭ. ‘An example of a semiprimitive semigroup algebra.’ Semigroup Forum 26 (1983), 225–228.
C. Rickart. ‘The uniqueneśs of norm problem in Banach algebras.’ Ann. Math. 51 (1950), 615–628.
A.V. Rukolaĭne. ‘The centre of the semigroup algebra of a finite inverse semigroup over the field of complex numbers.’ Rings and linear groups. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 75 (1978), 154–158. [In Russian].
A.V. Rukolaĭne. ‘Semigroup algebras of finite inverse semigroups over arbitrary fields.’ Modules and linear groups. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 103 (1980), 117–123. [In Russian].
M.L. Teply, E.G. Turman and A. Quesada. ‘On semisimple semigroup rings.’ Proc. Amer. Math. Soc. 79 (1980), 157–163.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company
About this chapter
Cite this chapter
Munn, W.D. (1987). A Class of Inverse Semigroup Algebras. In: Goberstein, S.M., Higgins, P.M. (eds) Semigroups and Their Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3839-7_14
Download citation
DOI: https://doi.org/10.1007/978-94-009-3839-7_14
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8209-9
Online ISBN: 978-94-009-3839-7
eBook Packages: Springer Book Archive