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Ideal Theory of the Abelian Group-Algebra

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Ideal Theory of the Abelian Group-Algebra IDEAL THEORY OF THE ABELIAN GROUP-ALGEBRA. ]~Y T. VENK&TARAYUDU, M.A. (University oŸ Madras.) Received September 30, 1937. (Communicated by Dr. R. Vaidyanathaswamy, ~t.A., D.SC.) 1. Introduction. la: is well known 1 that the group-algebra over a commutative corpus K defined by a finite group G is semi-simple, when the characteristic p of the corpus K is not a divisor of the order N of the group and that in this case, the group-algebra K [G] is the direct sum of simple algebras whose products in pairs ate all zero. Beyond these general properties, the structure of the group-algebra is not completely kuown even in the case wheu G is Abelian. For example, in the existing literature, the following questions have not been discussed :-- (1) In the decomposition of K [G] us the direct sum of simple algebras, the actual number of simple component algebras. (2) The radical of K [G] when p is a divisor of N. (3) The residue class ring of K [G] with respect to its radical when it exists. In the present paper, we consider the case when G is Abelian and we show that in this case K [G] is simply isomorphic with the residue class ring of the polynomial domain K [xi, xa, xr] with respect to the ideal (x~'--l, x~~-1, -x~, --1), where ni, na "n~ are the orders of a set of basis elements of the Abelian group G. From a study of this residue class ring, we deduce al1 the important structural properties of the Abelian group-aigebra K [G]. We show in particular that, when Ÿ is not a divisor of N, K [G] is the direct sum of n corpuses which mutually annull each other and that n depends on the ground corpus K. When K is the corpus of rational numbers, n is mŸ and is equal to// d i where di is the number of divisors of n i. When the ground corpus K is sufficiently extended n is maximum and is equal to N and in this case, the component corpuses are simply isomorphic with the ground corpus K. When p is a divisor of N, K [G] is the direct sum of m mutually annulling primary 1 For instance see Wedderburn, Lectures on Matrices, p. 167-8. 118 Ideal Theory of the A belian Group-A/ge~ra, 119 rings and when the ground corpus K is sufficiently extended, mis equal to the greatest block factor, prime to p, of N. If G1, G2, Gk ate the Sylow- sub-groups of G and ir pis the prime associated with the Sylow component G1 of G, we shall show that the residue class ring of K [G] with respect to its radical is simply isomorphic with the Abelian group-algebra K [G/G1], defined by the factor group G/G1. Throughout this paper, I follow the notation of Van der Waerden.~ K [xl, x2, . xr] denotes the ring of polynomials in xi, xz, xr with coefficients from K. The highest common factor of the t ideals mi, m,, tot is the least ideal containing aH elenlents of mi, 111z, tot and is denoted by (mi, taz, tot). The least common multiple of mi, taz,. mi is the ideal consisting of all elements common to mi, tas, tot and is denoted by [11tl, taz, lnt]. Ir is the iutersection of the aggregates rol, ma, " ni. German letters denote ideals and in particular p's denote prime ideals aud q's denote primary ideals. 2. The Abelian Group-Algebra K [G~. Let K be a eommutative corpus with characteristic p (p may be a prime of 0) and al, c,~, ah be a system of n elements linearly independent in relation to K. We take the ,~'s to be permutable with the elements of K and we denote the elements of K by Greek letters. Let A be the module of linear forros 3 al~,+az)~.o+" +a~)t n. We take the unir el-ement 1 of the corpus K as the unit operator, a Ir the module of linear forros Ais closed under a second rule of combi- nation, which may be called multiplication and which is associative and both ways distribntive with respect to addition, i.e., if Ais simultaneously a ring, we say that Ais an algebra over the corpus K of finite rank n. The algebra is therefore conlpletely determined, when the basis elements and the multiplication seheme aiai = lr~~ . a k (i, j, k =1,2,. n) (1) of the basis elements ate known. The only condition which the multipli- cation table (1) is subject to is, that ir should be consistent with the associ- ative law for the basis elements, namely, .,. (,~,,~,) = (a,.,~.) ., (r, s, ~ = 1, '2,. n) (2) = Moderne Algebra, Bd. I and II. These will be referred to in the sequel as W. I and W. II. 80r additive Abelian group with operators, having K as the operator domain. i.e., 1. a=aforeveryainA. W. II, p. ll0. 120 T. Venkatarayudu We notiee that the group relation is a simple case, in which the associative law (2) is true. We can therefore take the elements gl, g~, " " " gr of a group G as the basis elements of ah algebra over the corpus K. Here the constants of multiplication r~s are either 0 or 1. The algebra so formed by the elements of a group is known as the group-algebra and the rank of the algebra is the order of the group. In the present paper we take the case when G is commutative (or Abelian) and we accordingly call the algebra obtained by the elements of a finite Abelian group G, the A belian group-algebra K [G]. 3. The Isomorphism between the A belian Group-Algebra K (G) and the Resi,4ue Class Ring K [xl, x~. xr]/(x~" -- 1, x~ ~ -- 1, ...x~~ -- 1). We know fronl group theory, that every finite Abelian group G has a basis P~, P~ Pr of orders say n~, ~~.,, n~ such that the elements of G can be uuiquely expressed in the fo~m PŸ P~' 0<ti.<n/ i=1,2,. .r. K [G] consists therefore of the totality of the elements of the forro. ~Dft ][9t~ ir - t~x ~2 " " P, 0 < t i < 1~i, i == 1, ~, r at in K. Since the unit element 1 of K can be taken as the nnit operator, as stated before, K [G] has the modulus E, the identity element of G itself. Now consider the ring of polynomials in x~, x2, xr with coeffieients from K, namely the polynomial domain K [x~, x2,. xr]. A correspond- enee may be established between the elements of the Hng K [x~, x,, xr] and the elements of K [G] by which A --+ ,~E, and xi --+ ]Pi for i = 1, 2, r. Ttm correspondence is obviously a homomorphism and from the general theory of homomorphisms, the elements in K [x~, x2, . xr] which corres- pond to the zero element of K [G] forro ah ideal in K [x~, x.,, xr~. To prove that this ideal is m = (x~, -- 1, x~-" -- 1, x~", -- 1), we note in the first instance that the polynomials x 7 --1-+0 fori ~1,2,. .r and therefore any element of the ideal m corresponds to the zero element of K [G]. Since division by x~~- 1 is possible in K [xi, x~,. x~J, anypolynomial in K [x~, x2,.., xr] redueed mod mis a polynomial f (x 1, x.,,... Xr) of degrees less than na, n.,,.. "nr in xi, x~,'''Xr respectively. Then f = Z at ~'~ol ~"02 x," -+I at PI ~ P~' P,'~" , O < t i< n i ," i=:1,2,. .r .... (3) Ideal Theory of lhe A3elian Group-Algebra 121 Since the elements of G are linearly independent in relation to K the expression 2;' atP[ ~ Pi' " P~' summed for 0 < li < ni is zero if and only ir the coeffieients at ate all zero, in which case, the left-hand expression in (3) is also zero. Ir therefore follows that the only elements in K [x I, x2,- "x~] which correspond to the zero element of K [G] ate the elements of the ideal m. From the general theory of homomorphisms ir follows that I~ [c] ~5 K (x~, x~, .... x ~)/m. 4. Some General Results in Ideal Theory for Later Application. Let tt be a ring and ii ah ideal in it. Let ~ be the residue class ring ~/ii. If 11 is an ideal in t~, ~ = b/a is an ideal in ~ and Ÿ uniquely deter- mined by ~. Obviously ~ is the same if we replaee [j by (b, a). Conversely an ideal ~ in ~ is composed of a system of residue classes mod ii and the totality of the elements in the system of residue elasses forro an ideal b in ~ containing the ideal ii. tb is uniquely determined by b. Lemma l.O~If bis a divisor of ii, proof, t~ ~~ ~ ~ ~/~ and 0 in ~]~ --> ~ in ~ --+ b in t~ and conversely. Therefore by the general theory of homomorphisms ~/b _- ~/b. Corollarv. I[ [j is a prime ideal in lA, then ~ is a prime ideal in ~,, If tb is a faetorless prime ideal in t~, ~o also is ~ in ~. We assume the following :-- THEOREM 1.7 Ir t~ is a commutative ring with unir eiement and with finite basis for ideals and ir t~ [xi is the l"ing of polynomia]s in x with eoeffi- cients from ~, every ideal a in t~ [xi has a canonical basis in the forro (al1 , al~ , .
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