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Semigroup Gradings on Associative Rings

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Semigroup Gradings on Associative Rings View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Advances in Applied Mathematics 37 (2006) 153–161 www.elsevier.com/locate/yaama Semigroup gradings on associative rings Y.A. Bahturin a,b,∗,1,M.V.Zaicevb,2 a Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C5S7, Canada b Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow 119992, Russia Received 2 May 2005; accepted 23 June 2005 Available online 12 May 2006 Dedicated to Amitai Regev on this 65th birthday Abstract The main aim of this paper is the adaptation of the results of the papers [Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2) (2001) 677–698; Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Finite-dimensional graded simple algebras, preprint, and M.V. Za˘ıtsev, S.K. Se- gal, Finite gradings of simple Artinian rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 3, 2001, pp. 21–24, 77 (in Russian); translation in Moscow Univ. Math. Bull. 56 (3) (2001) 21–24] about the struc- ture of graded simple rings from the gradings by groups to the gradings by cancellative semigroups. © 2006 Elsevier Inc. All rights reserved. Keywords: Graded ring; Simple ring; Matrix algebra; Cancellation semigroup 1. Introduction Recently a number of classification-type results about the structure of graded ring and algebras have been obtained in the case where the grading object is a group [1–3,7]. Many of these results are no more valid if we allow the gradings by arbitrary semigroups. For example, in the case of groups there are only two basic types of gradings on the matrix algebra R = Mn(F ) (see [3]). Firstly, elementary gradings, where each matrix unit is homogeneous and all diagonal matrices have trivial grading. Secondly, fine gradings where each component is 1-dimensional, spanned * Corresponding author. E-mail addresses: [email protected] (Y.A. Bahturin), [email protected] (M.V. Zaicev). 1 This work was accomplished during the first author’s stay at Vanderbilt University in Spring semester 2005, supported by NSF Grant # DMS 0072307 (Professor Alexander Olshanskii). 2 The work is partially supported by SSC-1910.2003.1. 0196-8858/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.aam.2005.06.009 154 Y.A. Bahturin, M.V. Zaicev / Advances in Applied Mathematics 37 (2006) 153–161 by a non-degenerate matrix. Any other grading can be obtained from these two by applying tensor products of algebras and so called induced gradings of the tensor products of graded algebras. An obvious candidate for an alternative grading of Mn(F ) in the case of gradings by semigroups is the natural grading by the subsemigroup S ⊂ Mn(F ) of matrix units and 0: S ={Eij | 1 i, j n}∪{0}. Yet, as we will see in this paper, for a wide class of cancellative semigroups (see Definition 2) many results of the above mentioned papers remain true. It should be stressed that cancellative semigroups can be quite far from the groups. The condition that defines cancellative semigroups is necessary for a semigroup to be embeddable in a group but not sufficient [4]. Thus we cannot “automatically” assume that a ring graded by a cancellative semigroup is also graded by a larger group. But when we deal with graded simple ring there is a tool (Proposition 2) that allows us to reduce the grading by semigroups to the gradings by groups. 2. Notation and general results Let R be a ring and G a semigroup. A G-grading on R is an additive direct sum decomposition R = Rg such that RgRh ⊂ Rgh for any g,h ∈ G. (1) g∈G The subset Supp R ={g ∈ G | Rg = 0} is called the support of the grading (1). The grading is called finite if Supp R is a finite set. The elements in the component Rg are called homoge- ∈ = neousof degree g.Ifx Rg we will write deg x g. A subring S of R is called graded if = ∩ S g∈G(S Rg).IfG possesses the identity element 1 then R1 is called the identity compo- nent. The grading is called trivial if Rg = 0 for any 1 = g ∈ G. = → = Definition 1. A mapping ϕ : R g∈G Rg S g∈G Sg is called a homomorphism (iso- morphism)ofG-graded rings, G is a semigroup, if ϕ is an ordinary ring homomorphism (iso- morphism) and ϕ(Rg) ⊂ Sg,g∈ G. A simple remark is as follows. Lemma 1. Let R and S be two G-graded rings,G is a semigroup. Suppose G is a subsemigroup ⊂ = → = of G such that Supp R,Supp S G.Ifϕ : R g∈G Rg S g∈G Sg is a homomorphism (isomorphism) of G-graded rings then ϕ is also a homomorphism (isomorphism) of G-graded rings. By the way, if a ring R is graded by a subsemigroup G of a semigroup G then it can easily be turned into a G-graded ring if one keeps Rg for g ∈ G and additionally sets Rg = 0, for g ∈ G \ G. Definition 2. [4] A semigroup G is called cancellative if for any g,x,y ∈ G any one of the relations xg = yg or gx = gy implies x = y. From now on all semigroups will be cancellative. A well-known fact, which we prove for easier reference, is the following. Y.A. Bahturin, M.V. Zaicev / Advances in Applied Mathematics 37 (2006) 153–161 155 Lemma 2. If e is an idempotent of a (cancellative) semigroup G, that is, e2 = e, then e is the identity of G. Proof. Indeed, for any g ∈ G we have ge = ge2. Therefore, ge = (ge)e. By the cancellation property, g = ge. Similarly, g = eg and then e is the identity element of G. 2 If G has an identity element we will denote it by 1. Another simple remark is as follows. Lemma 3. If G is a cancellative semigroup and R is a G-graded ring with unit element I then I ∈ R1. = + Proof. For if I I1 g=1 Ig is the decomposition of I as the sum of homogeneous compo- nents then for any homogeneous element x ∈ Rh one has x = xI1 + xIg. g=1 By the cancellation property only one xIg can be non-zero. If this happens for some g then h = hg = hg2. Canceling h on the left we find that g is an idempotent. By Lemma 2, g must be the identity 1 of G. Hence, x = xI1 for any homogeneous x ∈ R. Similarly, x = I1x. It follows that I1 is the unit element of R so that I1 = I and thus I is homogeneous of degree 1. 2 Corollary 1. A ring with unit element cannot be graded by a cancellative semigroup without identity. A G-graded ring R is called graded simple if R has no nonzero proper graded ideals different from R itself. Proposition 1. Let R be a G-graded simple ring, G a cancellative semigroup with 1.IfR1 is nonzero then any g ∈ Supp R is invertible. Proof. Take g ∈ Supp R. Then Rg = 0. Consider the two-sided ideal P = RRgR.IfP = 0 = 2 = 2 = then Rg is a nonzero graded ideal of R, hence R Rg. But also Rg 0 and so R 0. This contradicts the graded simplicity of R. Otherwise, P = R. Any element in P is the sum of the = = = = ∈ = form u i aixibi where deg xi g,degai hi ,degbi ki , hi,ki G. Since P R, there is nonzero u of degree 1. Then one of the components aixibi will be of degree 1. In other words, higki = 1. Multiplying both sides by hi on the right we will have higkihi = hi . By cancellation, −1 g(kihi) = 1. Similarly, (kihi)g = 1 and g = kihi . Thus any element in Supp R is invertible, as claimed. 2 A quick corollary is the following. Corollary 2. Let G be a cancellative semigroup and R a G-graded simple ring with the unit element I . Then G has identity 1 and there is a subgroup G ⊂ G such that Supp R ⊂ G. This corollary shows that in the study of algebraic structure of R one can replace G, which is a semigroup, by G, which is a group. The following lemma in the case where G is a group is due to Cohen–Rowen [5]. 156 Y.A. Bahturin, M.V. Zaicev / Advances in Applied Mathematics 37 (2006) 153–161 Lemma 4. Let G be a cancellative semigroup, R a G-graded ring with a finite grading. If S = Supp R does not contain the identity element 1 of G and |S|=n then Rn+1 = 0. ∈ ∈ = Proof. Consider arbitrary homogeneous elements x1 Rg1 ,...,xn+1 Rgn+1 .Ifallh1 g1,h2 = g1g2,...,hn+1 = g1g2 ···gn+1 are pairwise distinct then at least one hi does not belong ··· ∈ = to S. It follows that x1 xi Rhi 0. If the chain h1,h2,...,hn+1 contains at most n distinct = + = elements then hi hj for some 1 i<j n 1. Then hi hihi . As before, we derive that = ··· = ··· ∈ = ∈ = hi 1. In this case gi+1 gj 1 and xi+1 xj R1 0 since 1 H Supp R.Wehave n+1 verified the equality x1 ···xn+1 = 0 for any homogeneous x1,...,xn+1, hence R = 0. 2 Proposition 2. Let R be a ring with a finite G-grading, G a cancellative semigroup.
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