Group Gradings on Full Matrix Rings

Journal of Algebra 220, 709᎐728Ž. 1999 Article ID jabr.1999.7897, available online at http:rrwww.idealibrary.com on

S. Dascalescu,˘˘ B. Ion, C. Nastasescu˘˘

Faculty of Mathematics, Uni¨ersity of Bucharest, Str. Academiei 14, View metadata, citation and similar papers at core.ac.uk brought to you by CORE RO-70109 Bucharest 1, Romania provided by Elsevier - Publisher Connector and

J. Rios Montes

Instituto de Matematicas, UNAM, Zona Comercial, Apartado 70-637 C.P. 04510, Mexico, D.F., Mexico

Communicated by Kent R. Fuller

Received May 3, 1999

We study G-gradings of the matrix ring MknŽ., k a field, and give a complete description of the gradings where all the elements ei, j are homogeneous, called good gradings. Among these, we determine the ones that are strong gradings or crossed products. If G is a finite cyclic group and k contains a primitive <

of 1, we show how all G-gradings of MknŽ.can be produced. In particular we give a precise description of all C22-gradings of MkŽ.and show that for algebraically closed k, any such grading is isomorphic to one of the two good gradings. ᮊ 1999 Academic Press

INTRODUCTION AND PRELIMINARIES

Let G be a group, k a field, and V a finite dimensional G-graded s , k-vector space. If n dimŽ.V , then End Ž.V Mkn Ž.has a structure of a G-graded k-algebra. Conversely, given a matrix ring MknŽ.and a group G, it is a challenging problem to describe the algebra G-gradings of MknŽ.. An interesting type of grading is one for which all the matrices ei, j are homogeneous elements, where ei, j is the matrix with 1 on the Ž.i, j -posi- tion, and 0 elsewhere. We call these good gradings, and it turns out that a grading is good if and only if it can be obtained from an n-dimensional G-graded vector space as above. We give several characterizations for

709 0021-8693r99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. 710 DASCALESCU˘˘ ET AL. gradings isomorphic to good gradings; in particular we show that any grading of MknŽ.by a torsion-free group is isomorphic to a good grading. Good gradings have appeared before in another setting, in the work of wx Green and MarcosŽ see 3, 4.Ž. , in which Mkn is viewed as a quotient of the path algebra of the quiver ⌫, where ⌫ is the complete graph on n points. Then good gradings arise from weight functions on ⌫. Our approach is different, and the only overlap with the results contained in the cited papers is Proposition 2.1. We describe in Section 2 all good gradings <

MknŽ.. In Section 3 we give a method for doing this in the case where s G Cm, a cyclic group of order m, and k contains a primitive mth root of 1ŽŽ in particular char k.does not divide m .Ž. If char k.divides m, the method does not work. However, we are able to produce a Cp-grading of ) MkpŽ.which is not good, for any field k of characteristic p 0. In Section 4 we give a precise description of all C22-gradings of the algebra MkŽ.. The methods are essentially different when charŽ.k s 2 or char Ž.k / 2. If charŽ.k / 2, we use the results of Section 3. If char Ž.k s 2, we give a different approach. We characterize all such gradings isomorphic to a good grading. In particular, if k is algebraically closed, any C22-grading of MkŽ. is isomorphic to a good grading, no matter what the characteristic of k is. We mention thatwx 5 presents a general open problem, posed by E. Zelmanov, asking to describe all semigroup gradings of a full matrix ring. If G is a multiplicative group with identity element 1, we say that a s [ k-algebra R is G-graded if R g g GgR is a direct sum of k-subspaces : g such that RRghR ghfor any g, h G. In this case R is called strongly s g y s graded if RRghR ghfor any g, h G Žor equivalently RRgg1 R1 for g any g G., and R is called a crossed product if every R g contains an g invertible element. A non-zero element r R g is called homogeneous of degree g; we write degŽ.r s g. If any non-zero homogeneous element is invertible, R is called a graded division ring. A right R-module M is a s [ graded module if M g g GgM , a direct sum of k-subspaces, such that : g MRhgM hgfor any g, h G. We can define the category gr-R of all G-graded right R-modules, where a morphism between two objects M and N in this category is a morphism f: M ª N of R-modules, such that : g fMŽ.ggN for any g G. Similarly we can introduce the category R-gr of G-graded left R-modules. We refer towx 6 for all definitions and basic properties of graded rings and modules. GRADINGS ON MATRIX RINGS 711

1. GRADINGS FROM ENDOMORPHISM RINGS

Let R be a G-graded ring, and V a right G-graded R-module. For any ␴ g G let

s g ¬ : g ENDŽ.V ␴ Ä4f EndŽ.V fVŽ.g V␴ g for any g G , s which is an additive subgroup of End RŽ.V . Note that ENDŽ.V ␴ ␴ ␴ Hom RygrŽŽ..V, V , where Ž.V is the G-graded right R-module which is ␴ s just V as an R-module and has the shifted grading Ž.Vg V␴ g for any g g G. Then the sum Ý␴ g G ENDŽ.V ␴ is direct, and we denote this by s [ ENDRŽ.V ␴ g G ENDŽ.V ␴ , which is a G-graded ring. A similar con- struction can be performed for left graded modules. If R s k with the trivial G-grading, then a right graded R-module is just a vector space V s [ with a G-grading; i.e., V g g GgV for some subspaces Ž.V ggg G. In this s s situation we denote ENDŽ.V ENDkk Ž.V and End Ž.V End Ž.V .If V s , has finite dimension n, then ENDŽ.V End Ž.V Mkn Ž., and this in- ¨ duces a G-grading on MkniŽ..If Ž .1F iF n is a basis of homogeneous ¨ s F F elements of V, say degŽ.iig for any 1 i n, let Ž.Ei, j 1F i, jF n be the ¨ s ␦ ¨ F F basis of EndŽ.V defined by Ei, jtŽ. t, ji for 1 i, j, t n. Clearly s y1 degŽ.Ei, jijgg . We have an algebra isomorphism between EndŽ.V and MkniŽ.by taking E , jito e , jnfor i, j, and in this way MkŽ.is endowed with a good G-grading. In fact any good G-grading can be produced as above. To see this, we first need the following.

LEMMA 1.1. Let us consider a good G-grading on MniŽ. k . Then deg Že , i . s s иии - 1, degŽ.ei, jideg Že , iq1 .Ždeg eiq1, iq2 .deg Žejy1, j . for i j and s y1 y1 иии y1 ) degŽ.ei, jideg Že y1, ii .degŽ.e y2, iy1 degŽ.ej, jq1 for i j. s Proof. Since ei, i is a homogeneous idempotent, we see that degŽ.ei, i s иии 1. The second relation follows from ei, jiee, iq1 iq1, iq2 ejy1, j for any - s i j. The third relation follows then from eei, jj, iie , i, which implies s y1 degŽ.ei, jjdeg Ž.e , i .

PROPOSITION 1.2. Let us consider a good G-grading on MnŽ. k . Then there ¨ ( exists a G-graded ector space V, such that the isomorphism EndŽ.V Mkn Ž. with respect to a homogeneous basis of V is an isomorphism of G-graded algebras. g s y1 Proof. We must find some g1,..., gniG such that degŽ.e , jijgg for any i, j. Lemma 1.1 shows that it is enough to check this for the pairs g y y1 s F Ž.i, j ÄŽ.Ž.Ž1, 2 , 2, 3 , . . . , n 1, n .4, i.e., ggiiq1 degŽ.ei, iq1 for any 1 F y s s иии i n 1. But clearly gni1, g degŽ.Že i, iq1 deg eiq1, iq2 .deg Ženy1, n . for any 1 F i F n y 1, is such a set of group elements. 712 DASCALESCU˘˘ ET AL.

A G-grading of the k-algebra MkniŽ.is good if all e , j’s are homogeneous elements. However, there exist gradings which are not good, but are isomorphic to good gradings, as the following example shows. s s s EXAMPLE 1.3. Let R S Mk22Ž.with the C Ä41, g -gradings defined by

k 00k R s , R s , 1 ž/0 kkg ž/0 aby adc S s a, b g k , S s c, d g k . 1 ½5½5ž/0 bdg ž/yd

Then the map

ab aq cbq d y a y c f : R ª S, f s ž/ž/žcd c dy c / is an isomorphism of C2-graded algebras. The grading of S is not good, since e1, 1 is not homogeneous, but S is isomorphic as a graded algebra to R, which has a good grading. We see that in the previous example, although the grading of S is not good, the element e1, 2 is homogeneous. A corollary of the following result shows that in general a grading of the algebra MknŽ.is isomorphic to a good grading whenever one of the ei, j’s is a homogeneous element.

THEOREM 1.4. Let R be the algebra MnŽ. k endowed with a G-grading such that there exists V g R-gr which is simple as an R-module. Then there ( exists an isomorphism of graded algebras R S, where S is MnŽ. k endowed with a certain good grading.

Proof. As a simple MknŽ.-module, V must have dimension n. Let ⌬ s ENDRŽ.V ,asaG-graded algebra with multiplication the inverse map composition, hence V is a G-graded right ⌬-module, so we may consider s BIENDRŽ.V END⌬ Ž.V ,aG-graded algebra with map composition as ⌬ s s multiplication. Since V is a simple R-module and ENDRŽ.V ( ⌬ End RŽ.V k,so is isomorphic to k with the trivial grading. This shows that BIENDRŽ.V is the endomorphism algebra of a G-graded k-vector space of dimension n; thus it is isomorphic to MknŽ.with a certain good G-grading. On the other hand, the graded version of the density theorem wx ␸ ª ␸ ¨ s ¨ 2, Proposition 2.4 shows that the map : R BIENDRŽ.V , Ž.Ž.r r for r g R, ¨ g V, is a surjective morphism of G-graded algebras. As s ␸ Ann RŽ.V 0, we see that is injective, hence an isomorphism, which ends the proof. GRADINGS ON MATRIX RINGS 713

COROLLARY 1.5. If G is a torsion-free group, then a G-grading of MnŽ. k is isomorphic to a good grading. Proof. Let V be a graded simple moduleŽ with respect to the given wx G-grading of MknŽ... Then 1, Theorem 3.2 shows that V is a simple R-module, and the result follows from Theorem 1.4.

COROLLARY 1.6. Let R be the algebra MnŽ. k endowed with a G-grading g such that the element ei, j is homogeneous for some i, j Ä41,...,n . Then ( there exists an isomorphism of graded algebras R S, where S is MnŽ. k endowed with a certain good grading. s Proof. Since ei, jiis a homogeneous element, V Re , j is a G-graded R-submodule of R. Clearly V is the set of the matrices with zero entries outside the jth column, so V is a simple R-module, and we apply Theo- rem 1.4.

EXAMPLE 1.7. There exist gradings isomorphic to good gradings, but where no ei, j is homogeneous. Let R be the C2-graded algebra from s Example 1.3 and let S MknŽ.with the grading

2 a y b y2 a q 2b S s a, b g k , 1 ½5ž/a y b ya q 2b a y 2b ya q 4b S s a, b g k . g ½5ž/a y b ya q 2b

Then the map

ab 2 a q c y 2b y d y2 a y c q 4b q 2 d f : RªS, f s , ž/ž/žcd aq c y b y d ya y c q 2b q 2 d / is an isomorphism of C2-graded algebras. However, none of the elements ei, j is homogeneous in S.

2. GOOD GRADINGS OF MknŽ.

In this section we will study the good G-gradings of the algebra MknŽ.. The algebra MknŽ.endowed with a good G-grading will be freely regarded Ž.up to an isomorphism as a graded algebra END ŽV .s EndŽV .for a G-graded vector space V, as described in Section 1. We start by counting these gradings.

PROPOSITION 2.1. There is a bijecti¨e correspondence between the set of all y ª good G-gradings on MnŽ. k and the set of all maps f:Ä4 1,2,...,n 1 G, 714 DASCALESCU˘˘ ET AL. such that to a good G-grading we associate the map defined by fŽ. i s F F y degŽ.ei, iq1 for any 1 i n 1.

Proof. Lemma 1.1 shows that, to define a good G-grading on MknŽ.,it is enough to assign some degrees to the elements e1, 2, e 2, 3,...,eny1, n. The inverse of the correspondence mentioned in the statement takes a map y ª f:Ä4 1,2,...,n 1 G to the G-grading of MknŽ.such that

s degŽ.ei, i 1, s q иии y degŽ.ei, j fifi Ž.Ž.1 fj Ž1, . and

s y y1 y y1 иии y1 degŽ.ej, i fj Ž1 .fj Ž2 .fi Ž. for any 1 F i - j F n. <

We will describe now the good G-gradings making MknŽ.a strongly gradedŽ. respectively, a crossed-product algebra. ( PROPOSITION 2.3. Let us consider the algebra EndŽ.V Mn Ž. k with a s F F y s good G-grading such that degŽ.ei, iq1 hi for 1 i n 1, where V [ ¨ ¨ g g GgV is a graded ector space. The following assertions are equi alent:

Ž.i Mn Ž k . is a strongly graded algebra. / g Ž.ii Vg 0 for any g G.

Ž.iii All the elements of G appear in the sequence 1, h112, hh,..., иии hh12 hny1. Proof. We recall that V is an object of gr-k, where k is regarded as a G-graded algebra with the trivial grading. Then ENDŽ.V s End Ž.V is strongly graded if and only if V weakly divides Ž.␴ V for any ␴ g G Žby wx6. . This means that V is isomorphic to a graded submodule of a finite ␴ / direct sum of copies of Ž.V in gr-k, and it is clearly equivalent to Vg 0 g for any g G.If g1,..., gn are the degrees of the elements of the basis of ( V which induces the isomorphism EndŽ.V Mkn Ž., then

s y1 s y1 s y1 h112gg , h 223gg ,..., hny1 ggny1 n ; thus

s y1 s y1 y1 s y1 y1 иии y1 g211h g , g 3211h h g ,..., gnnh y1 hny211h g . GRADINGS ON MATRIX RINGS 715

/ g Then Vg 0 for any g G if and only if all the elements of G appear in the sequence g12, g ,..., gn. Since s y1 s y1 y1 s y1 y1 иии y1 g211h g , g 3211h h g ,..., gnnh y1 hny211h g this is equivalent toŽ. iii in the statement.

COROLLARY 2.4. If MnŽ. k has a good G-grading making it a strongly graded algebra, then<< G F n.

COROLLARY 2.5. Let<< G s m F n. Then the number of good gradings on

MnŽ. k making it a strongly graded algebra is

y ny1 iq1 m ny1 mn 1 q Ž.m y 1 yy Ž.Ž.1 m y i Ý ž/i is1, my1

iq1 m y 1 ny1 yyŽ.1 Žm y i y 1. . Ý ž/i is1, my2 s s s иии Proof. Let x11212h , x hh,..., xny112hh hny1. Clearly the y Ž.n 1 -tuples Žh12, h ,...,hny112 .Žand x , x ,..., xny1 .uniquely determine each other, so we have to count the number of maps f:Ä4 1,2,...,n y 1 ª y : q G such that G Ä41 ImŽ.f . This is N12N , where N 1Žrespectively, y ª y N2 . is the number of surjective maps f:1,2,...,Ä4Än 1 G 14Ž respectively f:Ä4 1,2,...,n y 1 ª G.. A classical combinatorial fact shows that y iq1 m N s mn 1 yyŽ.1 1 Ý ž/i is1, my1 and

ny1 iq1 m y 1 ny1 N s Ž.m y 1 yy Ž.1 Žm y i y 1, . 2 Ý ž/i is1, my2 which ends the proof. ( PROPOSITION 2.6. Let EndŽ.V Mn Ž. k with a good G-grading, where s [ ¨ ¨ V g g GgV is a graded ector space. The following assertions are equi a- lent:

Ž.i Mn Ž k . is a crossed product. s <<и g Ž.ii dim ŽV . G dimŽ.Vg for any g G. и <

Proof. ENDŽ.V s End Ž.V is a crossed product if and only if EndŽ.V ␴ s ␴ ␴ g Hom grykŽŽ..V, V contains an invertible element for any G, which means that V ( Ž.␴ V as k-graded modules. This is equivalent to s ␴ g m dimŽ.Vg dim ŽV␴ g .for all , g G, which is just Ž. ii . Thus Ž.Ž. i ii . 716 DASCALESCU˘˘ ET AL.

~~« s ClearlyŽ. i Žiii . . Suppose now that Ž iii . holds. As End ŽV .1 s [ s 2 End grykgŽ.V g GgEndŽ.V , we find dimŽ End Ž..V 1 Ýg g Gg Ždim ŽV ... Then~~

~~2 2 <~~

~~᎐ g and the Cauchy Schwarz inequality shows that all dimŽ.Vg , g G, must s <<и g m be equal. Thus dimŽ.V G dimŽ.Vg for any g G. Clearly Ž. iii Ž.iv .~~

COROLLARY 2.7. Any two crossed-product structures on MnŽ. k which are good G-gradings are isomorphic as graded algebras. Proof. The two graded algebras are isomorphic to EndŽ.ŽV respectively, EndŽ..W , where dim Ž.V s dim ŽW .s n for any g g G. Therefore gg<

As examples, we give a description of all good C22-gradings on MkŽ. and all good C23-gradings on MkŽ.. s EXAMPLE 2.8. Let R Mk2Ž., k an arbitrary field. Then a good C2-grading of R is of one of the following two types: s s Ž.i the trivial grading, R12MkŽ., R g 0; Ž.s Ž.k 00s Ž.k ii R1 0 kk, R g 0 . s EXAMPLE 2.9. Let R Mk3Ž., k an arbitrary field. Then a good C2-grading of R is of one of the following types:

s s Ž.i the trivial grading, R12MkŽ., R g 0; kk000k s s Ž.ii R1 kk000, R g k ; ž/00kkk ž/0 k 00 0kk s s Ž.iii R1 0 kk, R g k00; ž/0 kk ž/ k00 k 0 k 0 k 0 s s Ž.iv R1 0 k 0 , R g k 0 k . ž/k 0 k ž/0 k 0 GRADINGS ON MATRIX RINGS 717

Using Propositions 2.3 and 2.6, we see that the examplesŽ.Ž ii , iii . , and Ž iv . are strongly graded rings, but they are not crossed products. ExampleŽ. ii , due to E. Dade, was known as a strongly graded ring that is not a crossed productŽ seewx 7. . The same example also appears inwx 8, p. 131 .

~~3. GRADINGS OVER CYCLIC GROUPS~~

s Let m be a positive integer and let Cm ²:g be the cyclic group of order m. We assume that a primitive mth root of unity ␰ exists in k Žin particular this implies that the characteristic of k does not divide m.. s ¨ THEOREM 3.1. Let R MknŽ., where k is a field containing a primiti e ␰ mth root of unity . Then a Cm-grading of R is of the form

~~s q ␰yi y1 q ␰y2 i 2 y2 q иии R g i Ä A XAX X AX q␰yŽ my1.i my1 ymq1 ¬ g X AX A MknŽ.4 ,~~

~~g m g where X GLnnŽ. k is such that X kI .~~

Proof. Let R s [ g R i be a C -grading of R s MkŽ.. Define the i Z m gm n i map ⌿: R ª R by ⌿Ž.A s Ý g ␰ A i for any A g R. We obviously i Z m g have that ⌿ is a linear map. Moreover, for any A, B g R we have that

~~⌿ ⌿ s ␰ i ␰ j Ž.Ž.A B ÝÝAggi B j ž/ž/g g i Z mmj Z s ␰ iqj Ý Aggi B j g i, j Z m s ␰ s ÝÝ Aggi B j g q s s Z m i j s~~

~~s s ␰ s Ý Ž.AB g g s Z m s ⌿Ž.AB ,~~

showing that ⌿ is an algebra morphism. Moreover, for any j we have that j ji m ⌿ Ž.A s Ý g ␰ A i for any A g R. In particular ⌿ s Id, and ⌿ is an i Z m g algebra automorphism of R. By the Skolem᎐Noether theorem, an algebra ⌿ ⌿ s y1 g automorphism of MknŽ.is of the form ŽA . XAX for any A g ⌿ m s MknnŽ., where X GLŽ. k . In order to have Id, we must require m g s the condition X ZMŽŽ..nnn k kI , I the identity matrix. 718 DASCALESCU˘˘ ET AL.

~~It is possible to recover the grading from the automorphism ⌿. Indeed, g g let j Zm and A R. Multiply the equations~~

s ⌿ s ␰ i A ÝÝAggi , Ž.A A i ,..., g g i Z mmi Z ⌿ my1 s ␰ Žmy1.i Ž.A Ý Ag i g i Z m by 1, ␰yj, ␰y2 j,...,␰yŽ my1. j, respectively, and then add the obtained equations. We find that

~~q ␰yj ⌿ q иии q␰yŽ my1. j⌿ my1 s A Ž.A Ž.A mAg j ; therefore~~

1 s q ␰yj ⌿ q иии q␰yŽmy1. jm⌿ y1 A j Ž.A Ž.A Ž.A g m 1 s Ž.A q ␰yi XAXy1 q ␰y2 i XAX2 y2 q иии q␰yŽ my1.imXAXy1 ymq1 , m which ends the proof. g Remark 3.2. The proof of Theorem 3.1 shows that, for any A MknŽ., the homogeneous components of A in the grading defined by the matrix X as in the statement are

1 yi y1 y2 i 2 y2 yŽ my1.imy1 ymq1 A i s Ž.A q ␰ XAX q ␰ XAX q иии q␰ XAX g m for any 0 F i F m y 1. If the characteristic of k divides m, we cannot proceed in the same way for describing Cmn-gradings of MkŽ.since m is not invertible in R. Nevertheless, we are able to produce an example of a Cp-grading of MppŽ.Z which is not a good grading. G s PROPOSITION 3.3. Let p 3 be a prime number, R MppŽ.Z and g s s a, b R, a Ž.ai, j 1F i, jF pi, b Ž.b , j 1F i, jF p, where

~~s ␦ q ␦ y␦ q ␦ F F ai, jiq1, ji, pjŽ.,1 j,2 , for all 1 i, j p i y 1 b s , for all 1 F i, j F p i, j ž/j y 1 GRADINGS ON MATRIX RINGS 719~~

~~␦ s wx py1 i Ž.i, jpidenotes Kronecker’s delta . Then K Z a is a field, the sum Ý s0 Kb is direct, and~~

R s K [ Kb [ иии [ Kb py1 ¨ is a Cp-graded di ision ring structure on R. In particular, this grading is not isomorphic to a good grading. s p y q g wx Proof. Let PXŽ. X X 1 Z p X . It is well known that P is irreducible and it is the minimal polynomial of a, which is written in the s wx p Jordan canonical form. So, K Z p a is a field and it has p elements. s The matrix b has zero entries above the diagonal and bi, i 1, for i s 1, p; thus the minimal polynomial of b is Ž.X y 1 pps X y 1, and b is invertible. An easy computation shows that s q ab baŽ.Ip , so Kb s bK and b f K. Now everything follows if we show that the sum

K q Kb q иии qKb py1 F F y ␣ : is direct. We prove by induction that for any 0 j p 1, Ž.iis0, j K ␣ i s ␣ s F F such that Ý0 F iF jiŽ.2b 0, we have i 0, for every 0 i j. If j s 0, there is nothing to prove. If j s 1 and ␣ q ␣ s 01Ž.2b 0 ␣ sy␣ ␣ / ␣ sy y1␣␣y1 then 1011Ž.2b .If 0, then is invertible and b 2 10. g ␣ s ␣ s This means that b K, a contradiction. So 100, implying 0. If 1 - j - p y 1 and

~~jq1 ␣ i s Ý i Ž.2b 0,Ž. 1 is0 then multiplying this relation by 2b,2Ž.b 2,...,Ž. 2b py1 and adding them, we obtain~~

~~jq1 ␣ q q иии q py1 s Ý ipž/I Ž.2b Ž.2b 0. ž/is0 q q иии q py1 / jq1 ␣ g Because 1 2b Ž.2b 0 and Ýis0 i K we obtain~~

~~jq1 ␣ s Ý i 0.Ž. 2 is0 720 DASCALESCU˘˘ ET AL.~~

~~SubtractingŽ. 2 from Ž. 1 we get~~

~~jq1 ␣ i y s Ý ipž/Ž.2b I 0 is0 and since 2b y 1 is invertible~~

~~jq1 ␣ iy1 q иии q s Ý ipž/Ž.2b I 0, is0 which means that~~

~~jq1 jq1 ␣ q ␣ q иии q␣ j s ÝÝiiŽ.2b jq1 Ž.2b 0 is1 ž/is2~~

␣ s ␣ s and jq1 0 by the induction hypothesis jq1 0. NowŽ. 1 shows that ␣ s s q i 0 for every i 0, j 1. Finally we remark that a good Cpp-grading on MkŽ.cannot be a graded division ring, since the elements ei, j are homogeneous, but not invertible. This shows that our grading is not isomorphic to a good grading.

Remark 3.4. We can also produce a C2-graded division ring structure Ž. s Ž. s Ž.11 s on M22Z . Indeed, let R M22Z , and the matrices A 10 and B Ž.11 s Ä4Ä2 s 4 01 . Then R120, I , A, A and R g 0, B, AB, BA define a C2-grading on R. Moreover, this is clearly a graded division ring structure. If k is a field of characteristic p ) 0, since MkŽ., k m M ŽZ ., the p Z p pp Cpp-grading of M Ž.Z pgiven in Proposition 3.3 and Remark 3.4 extends to a Cpp-grading of MkŽ., which is not a good grading. Remark 3.5. We can obtain other graded division ring structures on s g y matrix rings in the following way. Let R MknŽ., and let S R mod theŽ. unique type of simple R-module. Assume that R has a G-grading such that S is not gradable. Then let ⌺ be a graded simple R-module. Since R is simple artinian and ⌺ is finite dimensional, we have ⌺ , S t as R-modules, for some integer t. Since S is not gradable, we have t ) 1. , t , ⌺ s ⌺ ⌺ Then MktRŽ.End ŽS .Ž.Ž.End REND R. Since is a simple object in the category R y gr,sois⌺Ž.␴ for any ␴ g G. Thus any element ⌺ ⌺ of ENDRŽ.␴ is either zero or invertible, showing that ENDR Ž.is a graded division ring. This transfers to a graded division ring structure on MktŽ.. GRADINGS ON MATRIX RINGS 721

~~4. C22-GRADINGS OF MkŽ.~~

The purpose of this section is to describe C22-gradings of MkŽ..We start with the situation where charŽ.k / 2, when we use the method developed in Section 3. We are able to give a very precise description in this case. / s THEOREM 4.1. Let k be a field with charŽ.k 2, and let R Mk2 Ž.. Then a C2-grading of the k-algebra R is of one of the following three types:

~~q ¨ R s au bu u, ¨ g k , 1 ½5ž/cu yau q ¨~~

~~¡ cb ¦ y ␦ y ␥␦ ~¥2 a 2 a R s ␥ , ␦ g k , g cb ¢§␥␦q ␥ 02 a 2 a where a, b, c g k, a / 0, and a2 q bc / 0;~~

Ž.ii ¨ ␥ b␦ R s bu u, ¨ g k , R s ␥ , ␦ g k , 1 ½5ž/cu ¨ g ½5ž/yc␦ y␥ where b, c g k y Ä40; ¨ s s Ž.iii the tri ial grading R12Mk Ž., R g 0. g / Proof. We start by finding all matrices X Mk2Ž., X 0, such that 2 g s Ž.ab 2 s ␣ ␣ g X kI22. Let X cd . Then X I for some k if and only if

~~a2 q bc s ␣ , d2 q bc s ␣ , baŽ.q d s 0, ca Ž.q d s 0. Ž.3~~

If a q d / 0, then b s c s 0, and a s d.If a q d s 0, then d sya and 2 q / s ␤ 0 a bc 0. Thus there are two types of matrix solutions: X Ž.0 ␤ , with ␤ g y Ä4 s Ž.ab g 2 q / k 0 , and X c ya , with a, b, c k, a bc 0. For the first type, we obtain the trivial grading, since

~~s q y1 ¬ g s ¬ g s R122Ä4A XAX A MkŽ. Ä42 A A MkŽ.Mk2 Ž.. 722 DASCALESCU˘˘ ET AL.~~

~~s Ž.ab 2 q / sgŽ.xy Ž. Now let X c ya , with a bc 0. If A zt Mk2 , then the homogeneous components of A in the C2-grading associated to X are~~

~~1 A s Ž.A q XAXy1 1 2 1 s 2Ž.a2 q bc~~

~~2 a2 q bc x q acy q abz q bct abx q bcy q b2 z y abt = Ž. ž/acx q cy22q bcz y act bcx y acy y abz q Ž.2 a q bc t 1 A s Ž.A y XAXy1 g 2 1 s 2Ž.a2 q bc~~

~~bcx y acy y abz y bct yabx q Ž.2 a2 q bc y y b2 z q abt = . ž/yacx y cy22q Ž.2 a q bc z q act ybcx q acy q abz q bct~~

~~We distinguish two possibilities. If a / 0, denote~~

~~u s Ž.ax q cy q bz y at r2Ž.Ža2 q bc , ¨ s x q t.r2.~~

~~s Ž.au q ¨ bu ¨ Then A1 cu yau q ¨ , and u, can take any values in k, since the Ž.acbya matrix100 1 has rank 2. Thus~~

~~q ¨ R s au bu u, ¨ g k . 1 ½5ž/cu yau q ¨~~

~~On the other hand, denoting~~

~~␥ syŽ.acx y c2 y q Ž.2 a2 q bc z q act r2 Ž.a2 q bc , ␦ syŽ.abx q Ž.2 a2 q bc y y b2 z q abt r2 Ž.a2 q bc we have~~

cb y ␦ y ␥␦ 2 a 2 a A s g cb ␥␦q ␥ 02 a 2 a GRADINGS ON MATRIX RINGS 723 and the matrix yac yc2 Ž.2 a2 q bc ac ž/yab Ž.2 a22q bc ybab has rank 2, so ¡ cb ¦ y ␦ y ␥␦ ~¥2 a 2 a R s ␥ , ␦ g k , g cb ¢§␥␦q ␥ 02 a 2 a showing that the grading is of typeŽ. i . s / s Ž.¨ bu s Žq .r If a 0, then bc 0. In this case A1 cu ¨ , where u cy bz 2bc and ¨ s Ž.x q t r2 run through the elements of k. Then denoting ␥ s Ž.x y t r2, ␦ s Žcy y bz .r2bc, we see that ␥ b␦ A s ␥ , ␦ g k , g ½5ž/yc␦ y␥ and this gives a grading of typeŽ. ii . / COROLLARY 4.2. AC22-algebra grading of MŽ. k , char Ž.k 2, different from the tri¨ial grading, is a crossed product.

Proof. It is enough to show that for any grading of typeŽ. i or Ž ii . , R g contains an invertible element. But this clearly follows from the fact that m␦ 22q n␦␥ q p␥ s 0 for any ␥, ␦ g k if and only if m s n s p s 0.

~~In the next two propositions we describe which of the C2-gradings of Mk2Ž.is isomorphic to a good grading.~~

~~PROPOSITION 4.3. Let b, c g k y Ä40.Then the grading~~

¨ ␥ b␦ R s bu u, ¨ g k , R s ␥ , ␦ g k 1 ½5ž/cu ¨ g ½5ž/yc␦ y␥ of M2Ž. k is isomorphic to a good grading if and only if bc is a square in k. s Ž. s Ž.k 00s Ž.k Proof. Let S Mk22with the trivial C -grading S10 kk, Sg 0 . If f: S ª R is an isomorphism of graded algebras, then there exists g s y1 g s pq Y GL2Ž. k with fA Ž . YAY for any A S. Let Y Ž.rs . Then for sg0 x A Ž.y 0 Sg we have

1 qsy y prx yqy22q px fAŽ.s YAYy1 sgR ps y qr ž/sy22y rx yqsy q prx g 724 DASCALESCU˘˘ ET AL. and this shows that bsyŽ 22y rx.Žq c yqy22q px. s 0 for any x, y g k. Thus br 22s cp and bs 22s cq . We obtain that bc s Ž.cp 2, a square in k. r Conversely, suppose that bc s d2 for some d g k. Let d 1 2c Y sgGLŽ. k . d 1 2 y 0b 2 sgx 0 Then for A Ž.0 y R1 we have

~~Ž.x q y r2 dy Ž.y x r2c YAYy1 s ž/dyŽ.y x r2bx Ž.q y r2~~

~~sg0 x and for A Ž.y 0 R g we have~~

~~Ž.Žby q 4cx r4d yby q 4cx .r4c YAYy1 s . ž/Ž.Ž.by y 4cx r4b y by q 4cx r4d These show that the map f: S ª R, fAŽ.s YAYy1, is an isomorphism of graded algebras.~~

~~PROPOSITION 4.4. Let a, b, c g k such that a / 0, a2 q bc / 0. Then the grading q ¨ R s au bu u, ¨ g k , 1 ½5ž/cu yau q ¨~~

¡ cb ¦ y ␦ y ␥␦ ~¥2 a 2 a R s ␥ , ␦ g k g cb ¢§␥␦q ␥ 02 a 2 a is isomorphic to a good grading if and only if a2 q bc is a square in k. Proof. Keeping the notation from the proof of Proposition 4.3, suppose that f: S ª R, fAŽ.s YAYy1, is an isomorphism of graded algebras. 0 x g Since fŽŽy 0 .. R g we find that cb qsy y prx syŽ.Ž. yqy22q px y sy 22y rx 2 a 2 a cb cb s q 22y syy p 22y rx ž/ž/2 a 2 a 2 a 2 a GRADINGS ON MATRIX RINGS 725 for any x, y g k. In particular

~~cb q2 q q 22y s s qs or c y 2 a y b s 0. 2 a 2 assž/ ž/~~

If bc s 0, then clearly a2 q bc is a square in k.If bc / 0, then, to have roots in k for the equation ct 2 y 2 at y b s 0 we need a2 q bc to be a square. Conversely, suppose that a2 q bc is a square in k. We first consider the / case where bc 0. Let t12, t be theŽ. distinct roots of the equation ct 2 y 2 at y b s 0, and let

~~tt X s 21. ž/11~~

~~If f: R ª S, fAŽ.s XAXy1, is the algebra isomorphism induced by X, then~~

~~0 x 1 tyy tx yty22q tx f sg12 12 S y 0 y y y x ytyq tx g ž/ž/t12t ž/12 and~~

~~b x 0 1 txy tyŽ. xy y f sg21c S , 0 y t y t 1 ž/ž/12 x y y ytxq ty 012 showing that f is an isomorphism of graded algebras. If b s c s 0, then R s S as graded algebras. If c s 0 and b / 0, then~~

~~11 X s 2 a 0 y 0b induces in a similar way a graded isomorphism between R and S. Similarly for c / 0, b s 0, and this ends the proof.~~

COROLLARY 4.5. If k is an algebraically closed field of characteristic not Ž. s Ž.k 0 2, then any C22-grading of the algebra M k is isomorphic either to R 10 k , s Ž.0 k s Ž. s R g k 0 or to R12Mk, R g 0. 726 DASCALESCU˘˘ ET AL.

We turn now to the characteristic-2 case. s THEOREM 4.6. Let R Mk2Ž., k a field of characteristic 2. Then a C2-grading of R is of one of the following two types: ¨ s s Ž.i the tri ial grading, R12MkŽ., R g 0;

~~x ␤ Ž.x q y Ž.ii R s x, y g k , 1 ½5ž/␣ Ž.x q yy ␣ x q ␤ yx R s x, y g k g ½5ž/y ␣ x q ␤ y~~

~~for some ␣, ␤ g k. g Proof. Let us consider a C2-grading of R. Then for any A, B R we have~~

~~s q Ž.AB 1 AB11 ABgg s q y y AB11 Ž.Ž.A AB1B 1 s q q AB AB11AB.~~

~~␸ ª ␸ s Let : R R, Ž.A A1. A straightforwardŽ. but tedious computation ␸ shows that the matrix of in the basis e11, e 12, e 21, e 22 is of the form~~

~~1 ␣␤0 ␤␥0 ␤ X s ␣ 0 ␥␣ 00 ␣␤1~~

for some ␣, ␤, ␥ g k Žto see this we let A and B run through elements of the basis in the previously displayed formula. . Since ␸ 2 s ␸, we must have X 2 s X, implying that either ␥ s 1, ␣ s ␤ s 0, or ␥ s 0. In the first case s ␸ s X I4; thus Id, and we find the trivial grading. Now let

~~1 ␣␤0 ␤ 00␤ X s ␣ 00␣ 00 ␣␤1 GRADINGS ON MATRIX RINGS 727~~

␣ ␤ g sgŽ.ab Ž. for , k.If A cd Mk2 , then a q b␣ q c␤␤Ž.a q d A s ␸ Ž.A s 1 ž/␣ Ž.a q ddq b␣ q c␤ x ␤ x q y s Ž. ž/␣ Ž.x q yy s ␣ q ␣ q ␤ s q ␣ q ␤ sgŽ.ab Ž. where x b c , y d b c . Also A cd Mk2 ; then s y Ag A A1 b␣ q c␤ b q a␤ q d␤␣x q ␤ yx ss, ž/c q a␣ q d␣ b␣ q c␤ ž/y ␣ x q ␤ y where x s b q a␤ q d␤, y s c q a␣ q d␣. Theorem 4.6 now allows us to obtain the same result for the charŽ.k s 2 case as for the charŽ.k / 2 case Ž cf. Corollary 4.2 . , albeit by completely different methods. s ¨ COROLLARY 4.7. If charŽ.k 2, then any non-tri ial C22-grading of MŽ. k is a crossed product. ␣ s ␤ s Proof. If 0, then clearly R g contains invertible elements. If at ␣ ␤ ␣ Ž.␣ 1 least one of and , say , is non-zero, then0 ␣ is an invertible element of R g . s s PROPOSITION 4.8. Let charŽ.k 2, and let R M2Ž. k with the grading x ␤ Ž.x q y R s x, y g k , 1 ½5ž/␣ Ž.x q yy ␣ x q ␤ yx R s x, y g k . g ½5ž/y ␣ x q ␤ y Then this grading is isomorphic to a good grading if and only if there exists t g k such that ␣ t 2 q t q ␤ s 0.

s pq Proof. We proceed as in the proof of Proposition 4.3. If X Ž.rs is an invertible matrix inducing an isomorphism of graded algebras between R and Mk22Ž.with the only non-trivial C -grading, then ␤ Ž.ps q rq s pq ␣ Ž.ps q qr s rs ␣ p2 q ␤ r 2 s pr ␣ q 2 q ␤s2 s qs. 728 DASCALESCU˘˘ ET AL.

If ␣, ␤ / 0, then p, q, r, s / 0ŽŽ.. since ps q qr s det X / 0 . Then ␣Ž.pp2 qq␤ s 0, and we take t s p.If ␣ s 0or␤ s 0, then clearly rr r there is t g k such that ␣ t 2 q t q ␤ s 0. Conversely, suppose that ␣ t 2 q t q ␤ s 0 for some t g k.If ␣ / 0, let g t12, t k be theŽ. distinct roots of this equation, and then the matrix tt X s 12 ž/11

~~␣ s s Ž.1 ␤ produces the required isomorphism. If 0, we take X 01 , which also induces an isomorphism as wanted.~~

~~COROLLARY 4.9. If charŽ.k s 2 and k is algebraically closed, then any~~

~~C22-grading of MŽ. k is isomorphic to a good grading.~~

~~ACKNOWLEDGMENTS~~

We thank the referee for very helpful comments and suggestions and for bringing to our attention paperswx 3 and wx 4 . We also thank Andrei Kelarev for very helpful discussions. Note added in proof. Using the results of sections 1 and 2, and counting the partitions of a n set with n elements in <

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