Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-07T23:20:12.292Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Jacobson radical of semigroup rings of commutative semigroups

Published online by Cambridge University Press:  24 October 2008

A. V. Kelarev
A. V. Kelarev
Affiliation:
Department of Mathematics and Mechanics, Ural State University, Lenina 51, Sverdlovsk, 620083, U.S.S.R.
Get access Rights & Permissions [Opens in a new window]

Extract

Many authors have considered the radicals of semigroup rings of commutative semigroups. A list of the papers pertaining to this field is contained in [4]. In [1] Amitsur proved that, for any associative ring R and for every free commutative semigroup S, the equalities B(RS) = B(R)S and L(RS) = L(R)S hold, where B is the Baer radical and L is the Levitsky radical. A natural problem which arises is to describe semigroup rings RS such that π(RS) = π(R)S, where π is one of the most important radicals. For the Baer and Levitsky radicals and commutative semigroups a complete solution of the above problem follows from theorems 2·8 and 3·1 of [15].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 3 , November 1990 , pp. 429 - 433
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Amitsur, S. A.. Radicals of polynomial rings. Canad. J. Math. 8 (1956), 355–361.CrossRefGoogle Scholar
[2]Clifford, A. H. and Preston, G. B.. The Algebraic Theory of Semigroups. Math. Surveys 7, vol. 1 (American Mathematical Society, 1961).Google Scholar
[3]Fotedar, G. L.. On semilattice rings. Indian J. Pure Appl. Math. 7 (1976), 616–619.Google Scholar
[4]Jespers, E. and Wauters, P.. A description of the Jacobson radical of semigroup rings of commutative semigroups. In Group and Semigroup Rings (Proceedings of Conference) (Johannesburg, 1986), pp. 43–89.CrossRefGoogle Scholar
[5]Jespers, E., Krempa, J. and Wauters, P.. The Brown–McCoy radical of semigroup rings of commutative cancellative semigroups. Glasgow Math. J. 26 (1985), 107–113.CrossRefGoogle Scholar
[6]Kelarev, A. V.. Radicals of semigroup algebras of commutative semigroups. In Proceedings XIX All-Union Algebraic Conference (Lvov, 1987), Abstracts, part II, p. 122.Google Scholar
[7]Kelarev, A. V.. On the Jacobson radical of semigroup algebras of commutative semigroups. In Proceedings III All-Union symposium on the Theory of Semigroups (Sverdlovsk, 1988), p. 34.Google Scholar
[8]Kelarev, A. V.. A description of the radicals of semigroup algebras of commutative semigroups. (To appear.)Google Scholar
[9]Kelarev, A. V.. Radicals and semigroup rings. Mat. Issled. 105 (1988), 81–92.Google Scholar
[10]Krempa, J.. Radicals of semigroup rings. Fund. Math. 85 (1974), 57–71.CrossRefGoogle Scholar
[11]Munn, W. D.. On Commutative semigroup algebras. Math. Proc. Cambridge Philos. Soc. 93 (1983), 237–246.CrossRefGoogle Scholar
[12]Munn, W. D.. On the Jacobson radical of certain commutative semigroup algebras. Math. Proc. Cambridge Philos. Soc. 96 (1984), 15–23.CrossRefGoogle Scholar
[13]Munn, W. D.. The algebra of a commutative semigroup over a commutative ring with unity. Proc. Roy. Soc. Edinburgh Sect. A 99 (1985), 387–398.CrossRefGoogle Scholar
[14]Munn, W. D.. The Jacobson radical of a band ring. Math. Proc. Cambridge Philos. Soc. 105 (1989), 277–283.CrossRefGoogle Scholar
[15]Okninski, J. and Wauters, P.. Radicals of semigroup rings of commutative semigroups. Math. Proc. Cambridge Philos. Soc. 99 (1986), 435–445.CrossRefGoogle Scholar
[16]Teply, M. L., Turman, E. G. and Quesada, A.. On semisimple semigroup rings. Proc. Amer. Math. Soc. 79 (1980), 157–163.CrossRefGoogle Scholar