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 multiplication table (1) is subject to is, that ir should be consistent with the associative 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

~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 correspond 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 determined 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~ , . alr ~, a21, a~2, a2r,,, ate, at 2, atrt) where ak~, ate 2, akr ~ are polynomials in ~ [xi whose highest coefficients forro an ideal Irk in t~ whieh is a proper divisior of bk-~ (k = 2, t). As a simple application of the above theorem we prove

~ Denotes simple isomorphism, ~ denotes homomorphism (multiple isomorphism). a E. Noether, " Abstrakter Aufbau der Ideal theori in alg. zahl. and Funktioneni korp6ren ", Math. Annalen, 1927, Bd. 96, p. 41. See Dr. R. Vaidyanathaswamy and T. Veakatarayudu, " Oa the canonical basis ~or ideals in a polynomial domain over a commutative ring with finite basis for ideals." To be published shortly. 122 T. Venkatarayudu

TttEORE~Ÿ 2. Ir mis ah ideal in t~ and f (x) a polynomial in ~ [xi with highest coefficient unity and ir the prime ideal divisors of m are all factorless prime ideals in t~ (i) the prime ideal divisors of the ideal [In, f (x)] in t~ [xi are of the form lp, • (x)~, where pis a prime ideal divisor of m and 4 (x) ah irreducible factor of f (x) mod p. (ii) The prime ideals lp, 4 (x)] ate factorless prime ideals. Pro@ The seeond part of the theorem is obvious. For in any ring, a~l ideal is a faetorless prime ideal ir the residue class ring wi~h respect to ir is a corpus an4 conversely. H_~re ~ (x) is an irreducible polynomial mod p, (f.e.) 4 (x) is ah irreducible p31yrtgmial with eoeffieients from the residue class ring t~/p. Sinee pis a factorless prime ideal in t~, the residue class ring t~{p is a corpus and therefore the residue class ring of ~ (x) with respect to lp, 4 (x)] is a corpus and lp, 4 (x)~ is a faetorless prime ideal. The first part of the theorem follows direetly from Theorem 1. For, let t0 be a prime ideal divisor of (m, f (x) and let the constant elements in p forro an ideal p in t~. p is a prime ideal in ~ otherwise p will not be a prime ideal in ~ [x] ; further pis a divisor of m for it eontains m. I-Ience p is a prime ideal divisor of m. pis therefore a factorless prime ideal in t~ by hypothesis and its proper divisor is the unir ideal with the basis element 1. In the canonical basis for ~, there should therefore be just one polynomial ~ (x) with highest eoefficient 1 by Theorem 1. Since ~ is a prime ideal, 4 (x) is an irreducible polynomial mod p, further 4 (x) is a factor of f (x) mod p, for f (x) is in tt. I-Ienee 4 (x) is an irreducible factor of f (x) mod p. tIence tt = [p, ~b (x)]. Asa further application of Theorem 1 we have TIt~:OR~~ 3. Ir llt is the produet of different factorless prime ideals of the forro p and f (x) has no repeated faetors mad p then [ni, f (x)] is the product of its different factorless prime ideal divisors. Pro@ Let Q be a primary ideal divisor of [ni, f (x)]. Let the constant elements in Q forman ideal 11 in tli. Since Q is primary, 11 must be primary in t~, further ii should be a divisor of in. I-Ience ii should be a primary ideal divisor of ni. But ni is only a produet of different factorless prime ideals and therefore ii can only be one of the factorless prime ideal divisors p of ni. As before in the canonical basis for Q, there should be just one polynomial ~b (x) with highest coefficient 1. Since Q is primary ~ (x) is a power of an irreducible polynomial mod p and further ~ (x) should be a factor of f (x) mod p. Bar by hypothesis f (x) has no repeated factor mod pand therefore ~ (x) can only be an irreducible factor off (X) mod p. Ideal Theory of [/te A6elia~z Grou-Mlge 123

Hence the primary idea] divisors of [In, f (x)] ate simply the factorless prime ideals lp, ~ (x)]. But two different factorless prime ideals are always without common faetors 8 and therefore Era. f (x)] is the produet 9 of the different faetorless prime ideals [P, ~ (x)]. TH~O~EM 4. In a ring ~ if

a ~=al, a2, . ar is the expression for ah ideal a as the produet of the ideals al, a2, ar which are without eommon faetors when taken in pairs, the residue class ring ~/a is the direet sum of r residue elass rings, whose produets in pairs ate all zero aud whieh ate resp~etively simp]y isomorphie with the residue class rings ~/ai ; i = 1, 2, r. Proof. The principle involved in obtaining the expression for the re~idue elas~ ring ~/a is no other than the application of the chinese remainder theorem to ideal mo:luli. We here gire only a sketeh of the proof? ~

If lJv = at a2, al/-1 al,,+ 1 ar v = 1, 2, g and bv = bv/a a and av = av/a every element ~ from t~/a can ba uniquely expressed in the forro b =b~ + ~~ + b~ bv belonging to ~v. Further av. ~v -~ 0 and therefore li~,. ~v = 0 /~ ff= v. The ring t~/a is the direet sum of the rings l~1, lJ2, br. From the general theory of homomorphisms (~v ~ ~ Sinee the sum and produet of two elements in t~/a are respeetively the sum of the corresponding sum and produet of the elements of the rings bl, b2, " " [i~ we have Corollary. Any ideal in ~/a is the direet sum of u~fiquely determined ideals in the rings lJx, 11~, b~.

5. The Radical of a Comm~#ative Ring. Ah element of a ring is said to be nilpotent if some power of ir is zero. Ah element a is said to be properly nilpotent if ax and xa ate nilpotent for every element x in the ring. The totality of the properly nilpotent elements forro a two-sided ideal whieh we eall the radical of the ring. In a eommutative ring, every nilpotent element is properly nilpotent and therefore the radical is formed by the totality of the nilpotent elements in the ring. If

s Two ideals are said to be without common factors ir their g. c. d. is the utait ideal. 9 W. II, p. 46. lo For details see W. II, p. 47. A4 F 124 T. Venkatarayudu zero is the only properly nilpotent element in a ring, we say that the ring has no radical. A ring without radical and with minimal condition for ideals is called semi-simple. Since the mŸ condition is always true in ah algebra, ah algebra is semi-simple when it has no radical. For commutative rings we have T~EOR~:M 5. ~~ If (0) = [ql, q~, " " " qt] is the decomposition of the zero ideal as the 1.c.m. of primary components, whose associated prime ideals Pi, P2, " Pi ate different, the radical of the ring is [PI, P, " " " Pt]. Pro@ I,et a be a nilpotent element and therefore aO = 0 for some O. .'. a p belongs to ~h, q~, " " " qt, .'. a belongs to PI, P2, " " Pi by the definition of primary ideals. Henee all nilpotent elements are in [Pi, P~, " " " Pi]. Conversely every element of lP1, P2, " " " P is nilpotent. For if ais an element of (Pi, P2, " " Pt) and ir ao~ belongs to qi then aO must belong to ql, q.,,' " Itt where p is the greatest of Pi, P2," " " Pi. ttence a o --= O. [P,, P~, " " " Ptl is therefore the radical of the ring.

6. The Decomposition of the Ideal m = {(x~, -- 1, x~~ -- 1, .x"~ -- 1)} in K [xl, x2, " x

Here two tases arise according as p [the characteristic of the corpus K] is a divisor of the produet nt, n2," "nr =N or not. We shall first consider the case when p ~ N. m 6 1. In this case the polynomiats xi ~ --1 llave no repeated factors and we llave T~I~;OR~:~ 6. m is the product of its different faetorless prime ideal divisors and ir the ground corpus K is sufficiently extended, the number of prime ideal divisors is equal to N. Pro@ In the polynomial domain K [x there is the g.e.d, process and every ideal in ir is a principal ideal and can be factored uniquely into product of prime power ideals. The prime ideal divisors of (x~* -- 1) are of the forro (~ wher.e ~ is an irreducible factor of x2* --1. The prime ideals (~ are all factorless prime ideals in K [x Since ni is not divisible by the characteristic of the corpus K, x~~ -- 1 has no repeated factors and therefore the ideal (x~~ -- 1) is the product of its different factorless prime ideal divisors ($1).

11 W. II, p. 156, Ex. 7. m p/N means pis a divisor of N. p• means p isnot a divisor of N. Ideal Theory of the ./l £ lian Gr ou p. M l ge3ra 125

Now eonsider the idea1 (xT, -- 1, x~' -- 1) in the polynomial domain K [xi, x2]. For this ideal, conditions of Theorems 2 and 3 are satisfied when wetake K Ex1] forO, x2 forx, x71--1fornI and x~'--lforf (x). If.6a is an irreducible factor of x~,- 1, and q~~ is an irreducible factor of xi' -- 1, 42 is obviously irreducible mod 41. Henee by Theorem 2, the prime ideal divisors of (x~' -- 1, xi" -- 1) are of the forro (q~l, ~b~) and they are factorless prime ideals. Therefore by Theorem 3, (x~ .... 1, x~' -- 1) is equal to the product of its different factorless prime ideal divisors. Thus by a successive application of Theorems 2 and 3 we find that the ideal (x~' -- l, x~' --1, . .x~"r--l)inK(xl, x.,,. .x~)is the produet of the different factorless prime ideals of the forro (~l, ~b,,, ~~) where ~i is ah irreducible factor of xT* -- 1 for i = 1, 2, r. If the given corpus K is the corpus of rational numbers x" - 1 = 17.6t (x) where t runs through the divisors of n and $t (x) is a cyelotomic polynomial, a3 In this case, the number of factoriess prime ideal divisors of mis equal to 17 d i where di is the number of divisors of n i. Ir the ground corpus K is such that the polynomials xi* -- 1, i = 1, 2,..-r are completely reducible into linear factors, the factorless prime ideal divisors of m are of the forro (xi--~1, x2--~.0,. .x~- ~r) where ~i is an nith root of unity (the unir element of the corpus K) and the number of prime ideal divisors is n~ n.o n~ = N. The theorem is thus complete. 6 ~. The discussion of the case, when pis a divisor of N is quite similar, but here unlike the previous case, the polynomials x7~- 1 may have repeated factors. If we put n =mp" (ro, p) =1 expanding (x m --1)#a by the Binomial theorem and using the fact that Ÿ is the characteristic of the corpus K we have 14

(x m _ l)#a __ x n ._ 1. Allalogous to Theorem 6 we prove TI~IEOREM 7. mis the product of m primary ideals, whose associated prime ideals are different and if tt,~e ground corpus K is sut¡ extended, mis equal to mi, m~, m, where ni ~ mip al ; (mi, p) =- ], i = 1, 2, r. (ii) The product of the different assoeiated prime ideals is the ideal

(xŸ - 1, x~"' - 1, x~r - 1).

~3 For the irreducibility of the polynomials ~t (x) see W. I, p. 158, ~4 W. I, p. 87. 126 T. Ven•atarayudu

Pro@ Proceeding as in Theorem 6, we find, that the prime ideal divisors of mare factorless prime ideals of the forro (~bl, ~b~, ~~.) where ~/'i is an irreducible factor of x~."* --1 (or x~* -- 1). It follows from the general ideal theory that the primary components of m ate without common factors when taken in pairs and that their 1.c.m. is equal to their product. The represen- tation is unique and the assoeiated prime ideals are the different ideals ('/Jl, ~2, "" ~br). When the ground corpus K is suffieiently extended so that the polynomials x~*--1 are completely reducible into linear factors, the factorless prime ideals are of the forro (xi -- ~l, x2 -- ~2,"" Xr -- s where r is an mith root of unity and the number of prime ideals is equal to the product mi m~. mr, (i.e.) the greatest block factor 15 of N prime to p. The second part of the theorem is obviou's.

6. 3. The residue class ring K [xl, x2,.." Xr]/m.--As ah immediate consequence of Theorems 4, 5, 6, 7 and Lernma 1 cor. we have the following two theorems.

THEOREIr 8. In the residue class ring K [xi, x2, .'. x~]/m (o) = ~1 ~, ... ~~ whe~ p Ÿ N = ~~ ~, .-., ,,~. (0) ---- It* ll= "'" q,n when p/N where lli and qi are ideals in the residue class ring K [xi, x 2 .. "Xr]/nt determined by the ideals Pi and (li in K [xl, xo, xr].

TH~:ORE~i 9. (1) The radical of the residue class ring K[x~, x2.. "Xr]/nt is (i) (o) whon ~ ;t- ~~ (ii) the residue class ring (xŸ~~ -- 1, x~" -- 1, ... x~', -- 1)]m when p/N (2) The residue class ring of K [xi, x~ Xr]/m with respect to its radical in (ii) is simply isomorphic with the residue class ring

K [x~, x~, ... x~]/(x~~ -- 1, x~, -- 1, ... x~, - 1).

7. The Properties of the Abelian Group-algebra K [(;].

7 1. Maximal and minimal condition for ideals.--If in every system of ideals in a ring, there is at least one maximal ideal (i.e.) one which is not con- tained in another ideal of the system and a minimal ideal (i.e.) one which contains no other ideal of the system, the ring is said to have maximal and minimal condition for ideals. Every algebra has maximal and minimal

1£ A factor A of Bis said to be a block factor if Ais prime to B/A, Ideal Theory of Ÿ Abelian Group-Algebra 127

condition for ideals, In particular, the Abelian group-algebra has maximal and minimal condition for ideals. 16 7. 2. Pr@erties zohen p J-N.--From the preceding paragraphs it fotlows that in this case the Abelian gronp-algebra is semi-simple and ~t is the direct sum of mutually annulling corpuses. When the ground corpus K is suffieiently extended the number of corpuses is equal to the order of the group. In this case ir is easy to see that the N corpus are simply isomorphic with the ground corpus K. For, the corpuses have been shown to be simply isonlorphie with I~ [x~, x,., ... x~]/ (x~ - G, x~ - ~~, ... x, - ~~) where Ii is an nith root of unity. If we choose the new variables Yl, Y~, " " •r suela that Yv =xv--[v for v =1, 2, ... r.

K[x~, x~, ... xr]/(xl- ~1, x2 -- ~2," " "/r - ~r) ~~-K [Yl, Yv "" "Yr]/(Ya, Y~, "'" Yr) I~. This corresponds to the fact that in a general semi-simple algebra, when the ground corpus K is sufficiently extended, the algebra is the direct sum of mutually annulling simple rings, each of which is simply isomorphic with the total matric algebra over a corpus, which is inverse isomorphic with the corpus of automorphisms of the eorresponding simple ring. 17 It may be noted that K [G] is never simple except in the trivial case, when the group G consists of the identical element only. The general group- algebra is also never simple, for ir always contains the two-sided ideal (g~ + g~ + "'" gN) generated by the sum of all elements of G. TI~EORE~ 10. Every ideal in K [G] is a principal ideal and also idempotent. Pro@ Let K [G] be the direct sum of the n corpuses K1, K~,...K n. By Theorem 4, cor. any ideal in K [G] is the direct sum of ideals in I(i (i = 1, 2,...n). But the Ki's ate corpuses and there ate only two ideals in a corpus, the zero ideal and the unir ideal. Therefore an ideal in K [G] is generated by ah element of the forro u~ + u~ + + u n where u i ig either

6 This fact may be directly proved fbr the residue elass ring K [xt, x2, XrJ[nl. The maxŸ condition for ideals is equivalent with the finiteness of the divisor chain of ideals which is again equivalent with finite basis for ideals in the ring. The rnaxi~al condition is trt~e in K [xl, x2, xr] and therefore it is also true in the residue class ring K [x~,x2,Xr]/nI. In a ring ~ with maximal condition for ideals, the necessary and sufficient condition for minimal condition for ideals in a residue class ring ~/~1 is that every prime ideal divisor of 1t in ~ is a Ÿ prime ideal. We showed that the prime ideal divisors of III ate factorless prime ideals. Hence the minimal condition for ideals in K [x~, x2, Xr][nt ~7 See W. II, p. 26 and pp. 153-54. 128 T. Venkatarayudu

the zero element or the unit element of Ki. It follows that the ideals in K [G] are principal and idempotent and that the number of ideals is 2 n. Since the zero ideal in K [GI is the product of different factorless prime ideals and every ideal is a divisor of the zero ideal, it follows that every ideal in K [G] is the product of different factorless prime ideals. TblEOREM 11. The units in K [GJ ate the elements in the direct sum

KI t @- K2' ~-[- .... -~- K n' where K i' consists of all elements of the corpus K i except its zero el The remaining elements in K [G] are al1 divisors of zero. Pro@ Since K iis a corpus, every element in K i' has an inverse. Let x = x:: + x2' + + xn' be any element in the direct sum K:' + K( +. + K~' and Yi' be the inverse of x i' in K i then y = YI' + Y2' + " " + Yn' is the inverse of x in K [G] and therefore every element in K:' + K.,' + ... +K'n is a unir. On the other hand, ira =ai' +a2' + ... +ah' is an element in K [GJ and some of them say arl, ar2,'"ar~ are the zero elen:ents of the corpuses Krl, Kr2," " "Kr, where r 1, r2, rk ate numbers in 1, 2, n and ir we take the element b = b 1 + b,~ + ... + bn in K [G] which is such that at least one of br: , br2, br~ is different from the corresponding arx, ar2, " " ar, and the remaiŸ b's are zero elements in the corresponding corpuses then, ab =0, a~O,b:#O. 7 3. Pr@erties of K [G] when p/N.--In this case, K [GJ is the direet sum of primary rings. We can show that there ate only a finite number of ideals in K [G] and that the units in K [Gj are the elements in the direct sum of the units of the eomponent primary rinos and the remaining elements in K [G] ate all divisors of zero. But the important properties of K [G] in this case ate connected with its radical.

TI-IEORE:VI ] 2. If G: is the Sylow component of G associated with the prime p (the characteristic of the corpus K) and if S:, $2, Sk ate a set of basis elements of G1, the radical of K [G] is the ideal (S: -E, S~ - E, ... S~ --E), where Eis the identity element of G. Pro@ A set of basis elements for G can be taken, for which the orders of the basis elements are prime powers. Such a set of basis elements is in fact the system of basis elements of the Sylow components of G. Therefore S:, $2, Sk can be taken asa part of the basis elements P:, P2, Pr say let Si=Pifor i = 1, 2, -.- k. By Theorem 9 the radical of K [GJ is the ideal (PI"' -- E, PŸ -- E, P,~' - E). i-Ience m 1 = m 2 .... mk -- 1 and m,+,, etc., ate all equal to the corresponding n's in which case Ÿ m -- E = pn_E__O, ttence the radical is (P: -E, P2-E,"" Pk--E). Ideal Theory of the ASelictn Group-AlKe£ 129

It is we11 known that the residue class ring of ah algebra with respect to its radical is semi-simple. In the case of the Abelian group-algebra we identify this semi-simple ring with another Abelian group-algebra. By Theorem 9 it follows that the residue class ring of K [G] with respect to its radical is simply isomorphic with K [xl, x.,, ... x~] / (x~", - 1, x~-2 - 1, ... x;, - 1) which is simply isomorphic with the Abelian group-algebra defined by ah Abeliau group which has a set of basis elements of orders mi, m2, "" my. We have therefore established the THEORF,~I. Ir the charaeteristic p of the ground corpus K is the prime associated with the Sylow component G1 of G, the residue class ring of K [G] with respect to its radical is K [G/G~] defined by the factor group G/Ga. I wish to thank Dr. R. Vaidyanathaswamy, M.A., D.SC., for his kind assistance in the preparation of this paper.