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The most goal of those lectures is first to in short survey the elemental con­ nection among the illustration concept of the symmetric team Sn and the speculation of symmetric features and moment to teach how combinatorial equipment that come up certainly within the thought of symmetric services bring about effective algorithms to precise a variety of prod­ ucts of representations of Sn when it comes to sums of irreducible representations.

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Gl. d. (R) ,;;;; 1 <=~ >I since (g~) is every submodule of a projective is projective *= every quotient module of an injective is injective <=~ every right ideal of R is projective. Such a ring is called (right) hereditary. The commutative hereditary domains are precisely the Dedekind domains (every ideal is invertible). Let R be a ring containing an infinite direct product of subrings. Then R is not hereditary. We will give two different proofs of this fact, both of which have additional interesting consequences.

For any A ~ w, let EA denote its characteristic function as an element of n;':oRj . Let w=U;oAj where Aj nA k =0for ji=-k Let F= {S ~P(w)IS'2 {A j 1i E w} and B, CES, Bi=- C ~ B n C is fmite}. F is inductive. Let So be a maximal element of F. Let {Bill ';;;;i';;;;n}5;So,Bji=-Bj if ii=-j. Assume Lf=lEBriE/. Then 45 HOMOLOGICAL DIMENSIONS OF MODULES n U;*jB; = Cj is finite. 'J· E I. Hence L BES EBR maps onto a direct sum J J J 0 modulo I. Let v be the natural map: M -7 M/I. Bj Define if>:(LSoEBR)/I-7M/I by if>(EAj ) = v(EAj),VjEw,if>(EB )= O,VBES o {A j /j E w}.

43. Let R be noetherian, ME NR the set of zero divisors on M. Then THEOREM Z(M) = = U ~OPj' Pj prime, (b) P prime 2 (0 : M) implies P 2 Pi (c) I f Z (M) ~ 30 of=- m E M, ml = O. 44. Let for some i. P be the set of all prime ideals of R. n pE pP is nil, and if R is noetherian it is nilpo ten t. (b) Unions and intersections of chains in P are again in P. 25 HOMOLOGICAL DIMENSIONS OF MODULES Definitions. (a) Let P be a prime ideal of R. Then let height P = ht P ;;;;. •• :::J Pn descending from P. (b) The Krull dimension of R, dim R = sup {ht PIP a prime ideal of R}.

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