The $$\pi $$ -Semisimplicity of Locally Inverse Semigroup Algebras
Tóm tắt
In this paper, we first characterize when a semigroup has completely 0-simple semigroup as its principal factors. Let R be a commutative ring with an identity, and let S be a locally inverse semigroup with the set of idempotents locally pseudofinite. Assume that the principal factors of S are all completely 0-simple. Then, we prove that the contracted semigroup algebra
$$R_0[S]$$
is
$$\pi $$
-semisimple if and only if the contracted semigroup algebras of all the principal factors of S are
$$\pi $$
-semisimple. Examples are provided to illustrate that the locally pseudofinite condition on the idempotent set of S cannot be removed. Notice that we extend the corresponding results on finite locally inverse semigroups.
Tài liệu tham khảo
Amitsur, S.A.: A general theory of radicals. II. Radicals in rings and bicategories. Am. J. Math. 76(1), 100–125 (1954)
Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroup. Mathematical Surveys, vol. 7. American Mathematical Society, Providence (1961)
Domanov, A.I.: On semisimplicity and identities of inverse semigroup algebras. Mat. Issled. 38(207), 123–137 (1976) (Russian)
Green, J.A.: On the structure of semigroups. Ann. Math. 54, 163–172 (1951)
Guo, X.J.: A note on locally inverse semigroup algebras. Int. J. Math. Math. Sci. 2008, 576061 (2008)
Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, New York (1995)
Ji, Y.D., Luo, Y.F.: Locally adequate semigroup algebras. Open Math. 14, 29–48 (2016)
Ji, Y.D., Luo, Y.F.: Semiprimitivity of orthodox semigroup algebras. Commun. Algebra 44(12), 5149–5162 (2017)
Kelarev, A.V.: Ring Constructions and Applications. World Scientific, New Jersey (2002)
Munn, W.D.: A class of contracted inverse semigroup rings. Proc. R. Soc. Edinb. 107A(1–2), 175–196 (1987)
Okninski, J.: Semigroup Algebras. M. Dekker, New York (1991)
Ponizovskĭ, I.S.: On the semiprimitivity of inverse semigroup algebras and on theorems by Domanov and Munn. Semigroup Forum 40, 181–185 (1990)
Rukolaĭne, A.V.: Semigroup algebras of finite inverse semigroups over arbitrary fields. J. Math. Sci. 24(4), 460–464 (1984)
Shojaee, B., Esslamzadeh, G.H., Pourabbas, A.: First order cohomology of \(\ell ^1\)-Munn algebras and certain semigroup algebras. Bull. Iran. Math. Soc. 35, 211–219 (2009)
Steinberg, B.: Möbius functions and semigroup representation theory. J. Comb. Theory A 113, 866–881 (2006)
Teply, M.L., Turman, E.G., Quesada, A.: On semisimple semigroup rings. Proc. Am. Math. Soc. 79, 157–163 (1980)
Weissglass, J.: Radicals of semigroup rings. Glasg. Math. J. 10, 85–93 (1969)