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The Brown Mccoy radical of semigroup rings of commutative cancellative semigroups

Published online by Cambridge University Press:  18 May 2009

E. Jespers ,
J. Krempa  and
P. Wauters
E. Jespers
Affiliation:
The University of Steluenbosch, Department of Mathematics, Stellenbosch, 7600–South Africa
J. Krempa
Affiliation:
University of Warsaw, Institute of Mathematics, University, Pkin, 00–901 Warsaw-Poland
P. Wauters
Affiliation:
Kathoueke Unrversitert Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B-3030 Leuven, Belgium
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We give a complete description of the Brown–McCoy radical of a semigroup ring R[S], where R is an arbitrary associative ring and S is a commutative cancellative semigroup; in particular we obtain the answer to a question of E. Puczyłowski stated in [11]

Throughout this note all rings R are associative with unity 1; all semigroups S are commutative and cancellative with unity. Note that the condition that R and S have a unity can be dropped (cf. [8]). The quotient group of S is denoted by Q(S). We say that S is torsion free (resp. has torsion free rank n) if Q(S) is torsion free (resp. has torsion free rank n). The Brown–McCoy radical (i.e. the upper radical determined by the class of all simple rings with unity) of a ring R is denoted by u(R). We refer to [2] for further detail on radicals and in particular on the Brown–McCoy radical.

First we state some well-known results and a preliminary lemma. Let R and T be rings with the same unity such that R ⊂ T. Then T is said to be a normalizing extension of R if T = Rx1+…+Rxn for certain elements x1, …, xn of T and Rxi = xiR for all i such that 1 ∨i∨n. If all xi are central in T, then we say that T is a central normalizing extension of R.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 2 , July 1985 , pp. 107 - 113
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Bergman, G. M., On Jacobson radicals of graded rings, preprint.Google Scholar
2.Divinsky, N. J., Rings and radicals (University ofToronto Press, London, 1965).Google Scholar
3.Jespers, E. and Puczyłowski, E., The Jacobson radical of semigroup rings of commutative cancellative semigroups, Comm. in Algebra. 11 (1984), 1115–1124.CrossRefGoogle Scholar
4.Jespers, E., Krempa, J. and Puczyłowski, E.,On radicals of graded rings, Comm. in Algebra, 10 (17) (1982), 1849–1854.CrossRefGoogle Scholar
5.Karpilovsky, G., The Jacobson radical of a commutative group ring, Arch. Math. 39 (1982), 428–430.CrossRefGoogle Scholar
6.Krempa, J., Radicals of semigroup rings, Fund. Math. 85 (1974), 54–71.CrossRefGoogle Scholar
7.Krempa, J., On radical properties of polynomial rings, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 545–548.Google Scholar
8.Krempa, J. and Sierpinska, A., The Jacobson radicalof certain group and semigroup rings, Bull. Acad. Polon. Sci. Sir. Sci. Math. Astronom. Phys. 26 (1978), 963–967.Google Scholar
9.Nastasescu, C. and Van Oystaeyen, F., Graded Ring Theory (North-Holland, 1982).Google Scholar
10.Passman, D. S., The algebraic structure of group rings Wiley Interscience, New York, 1977).Google Scholar
11.Puczyłowski, E. R., Behaviour of radical properties of rings under some algebraic constructions, to appear.Google Scholar
12.Schneider, H. and Weissglass, J., Group rings, semigroup rings and their radicals, J. of algebra 5 (1967), 1–15.CrossRefGoogle Scholar