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Andrei V. Kelarev FINITENESS CONDITIONS for SPECIAL BAND GRADED RINGS

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Andrei V. Kelarev FINITENESS CONDITIONS for SPECIAL BAND GRADED RINGS DEMONSTRATIO MAT IJ KM ATI CA Vol. XXVII No 1 1994 Andrei V. Kelarev FINITENESS CONDITIONS FOR SPECIAL BAND GRADED RINGS A number of chain (initeness conditions arc well-known classical concepts of ring theory. Properlies of this soi l have been investigated by many authors for various ring constructions, in particular, for semigroup rings (cf. [25]). The aim of the present paper is to carry out analogous investigations for special band graded rings, which arc closely related to semigroup rings. Namely, we shall deal with right Artinian, right Noetherian, semilocal, local, scalarly local, right perfect and scmiprimary rings. Let D be a band, i.e. a semigroup consisting of idempotents. An associa- tive ring R is said to be /¿-graded if R = ©¡,eö Rb-, the direct sum of additive groups, and RaR>, C Ra^ for any a, b £ B. A survey of papers concerning band graded rings was given in [10]. Here we mention only more recent ar- ticles [11]—[13], [19]. If each ring /?,(, has an identity 11,, and 101(, = la(, for any a,b then the ring II = Rb is called a special band graded ring, or a special X?-graded ring. This concept was introduced in [19]. In the case of commutative B special band graded rings were considered earlier in [1], [6], [27], [30] and other papers. Special band graded rings have found effective applications to the study of distributive and semiprimitive semigroup rings ([29], [9]), and to the investigations of the radicals of semigroup rings of CliiTord semigroups ([27]), orlhogroups ([28]), and commutative semigroups ([7], [14])- First, let us turn to semilocal special band graded rings. A ring R is said to be semilocal if the quotient ring R/.J(R) is right Artinian, where J(R) is the Jacobson radical of R. Semilocal semigroup rings were considered in [20], [22] and [31]. A band is called a semilattice (a rectangular band; a left zero band; a right zero band) if it satisfies the identity xy = yx (xyx = x; xy = x; XV = V)• Let be a semilattice. A band B is said to be a semilattice S of 172 A. V. Kclarcv rectangular bands II,, s G S, if B = Us(=s IIs, IIs fl //< = 0 whenever s ^ i, and //,//< C If st for any .s,t £ S. Each band is uniquely represented as a semilattice of rectangular bands (cf. [2], §4.2). This representation coincides with the splitting of JJ into classes by the equivalence relation a ~ b & aba = a and bah = b (cf. [19], §2). THEOREM 1. Let a band B be a semilallice S of rectangular bands II s. A special band graded ring 11 = lib is semilocal if and only if S is finite and all Rb are semilocal. A ring is said to be local (scalary local) if factoring out its Jacobson radical one gets a simple right Artinian ring (a division ring). Local and scalarly local semigroup rings were considered in [17], [20], [22] and [26]. Theorem 2. A spccial band gnu led ring R — ©¡,6B Rb is local (scalarly local) if and only if IS is a rectangular band and all Rb are local (scalary local). A ring K is said to be right 71-nilpotcnl if for any sequence xi, X2,... € K there exists a positive integer n such that ,rn.rn_1 ...a-j = 0. A ring R is right perfect (scmiprimary) if R/J(R) is right Artinian and J(R) is right T-nilpotent (nilpotcnt). Kiglit, perfect and scmiprimary semigroup rings were considered in [3], [-1], [23] and [16]. THEOREM 3. Let, a band B be a semilatlice S of rectangular bands Hs, and let R = Rb be a spccial band graded ring. Then R is right per- fect (semiprimary) if and. only if S is finite and all Rb are right perfect (semiprimary). Right Artinian and Noetlieriau semigroup rings were considered in [18], [32], [21], [15], [22], [5] and [2-1]. In particular, it is proved in [32] that if a semigroup ring RS is right Artinian, then S is finite and R is right Artinian. The converse implication is not valid. In [5] right Artinian commutative semigroup rings are described completely. We shall obtain analogous results for special band graded rings. Theorem 4. Let R = Rb be a special band graded ring. If R is right Artinian (right Noelhcrian), then B is finite and all Rb are right Artinian (right Noetherian). The following example shows that the converse is not true. Example 1. Let 7?. be field of real numbers, B = {a, b} a right zero band. Then the semigroup ring RB is a special band graded ring, RB = Ra + Rb. However, IIIJ is neither right Artinian, nor right Noetherian. In fact, evidently R contains an infinite chain of subrings ... C R-1 C Ro C R\ C Then it is routine to chcck that R\(a — b) C ^(a — 6) C ... is an Finileness conditions 173 infinite asccnding chain of light ideals of RB, and Ro(a-b) C R-i(a-b) C iZ_2(n. — 6) C ... is an in finite descending chain of right ideals of RB. If B is commutative, then we get COROLLARY 1. Let B be a scmilallice, R = Rb a special band graded ring. Then R is right Artinian (right Noetherian) if and only if B is finite and all Rb are right Artinian (right Noetherian). LEMMA 1 [19]. Let II be a reetangidar band, R = @be/i Rb a special band graded ring, If, the identity of Rb. Then x\hV — xy for any x,y G R, h G H. Proof. Let x = ^2beJIxb, y = X)6e//yb where xb,yb G Rb- Then x\hy = r x X X^a,6ef/' a lh b = Ea.be" XalalhlbXb = Za,f>e// a ^a 1&&6 = XIJ, because ahb = ab. The following key lemma is inspired by the result of [19]. LEMMA 2. Let a band B be a scmilallice S of reetangidar bands IIS, R = 06eB Rb a special band graded ring, U the identity of Rb. For each s G S choose an clement lis in II s and set Qs - Rb, Is = {® £ Qs | = 0}, / = Then I is an ideal of R contained in the Jacobson radical .1(11). The quotient ring R/I is a special S-graded ring, R/I = Fs, where Fs = Rh,. The ideal I and rings Fs do not depend on the choice of the elements hs. Besides,T^ = 0 for every s G S. A large part of the proof of Lemma 2 follows from several technical lemmas of [19]. For the sake of completeness we shall give a-self-contained proof. Proof. Take any .s G S and put II = //.,, h = hs. For any x,y G Qs, z G Is Lemma 1 and the definition of 7, yield xzy = xl^zl^y = 0. Therefore QshQs = 0, and so Ps = 0. Now we take any g G IIs and set h's = g, I's = {a; G Qs | igxlg = 0}. By Lemma 1 for any x G we get l^sl/i = l^lgxlglh = 0, whence I'3 C I3. Similarly, Is C I's. Thus Is and I do not depend on the choice of the elements hs. Let s,t e S, X G Is, y G Qt, h = hs, g - hst. Then ylg G QtQst Q Qst and 1 gX G QstQs C Qst• 1» view of Lemma 1 (lgx)(ylg) = lgxlhghy^g- Similarly, lg G Qsi, xlil!jhyln G Qst, and so by Lemma 1, lgxylg = lglhghxlhghylg- Given that \hgh = 1 h\g\h and x £ Is, we get lhgh^hgh = 0. Hence \gxylg = 0, that is xy £ Isl. Thus IsQt C Ist, which means that I is a right ideal in R. By duality, I is a left ideal of R, as well. Thus I is an ideal of R. The inclusion I C J(R) immediately follows from the description of the Jacobson radical of a special band graded ring which was obtained in [19]. 174 A. V. Kclarcv Here we give a separate proof. Every ring Is is radical, since = 0; and I = ®s€5/s is a semiiaUice graded ring. Theorem 3.2 of [1] shows that the radical class of the .lacobsou radical is closed under formation of semilattice graded rings. Therefore / is radical. Sincc J{ R) contains all the radical ideals of R, we get J(R) D I. Evidently, R/I = ®4gs Fs where Fs = (Qs + /)//. Furthermore, Fa Si Qa/(Qs n /) = Qs/Is. Hence the Fs do not depend on the choice of ha, s € S. Let s 6 S, g 6 //.,, x € Rg, h = ha. Then 1/tXl/i € Rh and it is clear that x —l/f-cl/i € h- Therefore Rg C Rh + Ia implying Rh + I3 = Qs- Besides i?/,il/, = 0, because Ihylh = y for every y € Rh. Hence Fa = Qa/Ia = (Rh + Is)/is = R)J(Rh n ls) s Rk. Let / be the natural liomomorphism of R onto R/I. Since — \g G Ia for every g,li £ IIs, then /(l/J = /(and it is easily seen that f(lg) coincides with the identity of Fs. Now take any s,t € S, g € II3, h € Ilf Then f(lg)f(lh) = /(l.,U) = /(lflA). This shows that f{lg)f{lh) is equal to the identity of Fsi- Tlicrefore R/I = (&S£SFs is a special band graded ring. LEMMA 3 ([31], Proposition 1.3). Let S be a semilattice, F = ©a£5 Fs a semilattice graded ring, Fs / J(FS) for every s 6 S.
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