# Download Communications in Algebra, volume 25, number 9, 1997 by Marcel Dekker, Inc PDF By Marcel Dekker, Inc

Goals and Scope. This magazine provides full-length articles thai replicate major advances in all components of present algebraic curiosity and job.

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The most goal of those lectures is first to in brief survey the basic con­ nection among the illustration idea of the symmetric workforce Sn and the idea of symmetric services and moment to teach how combinatorial tools that come up clearly within the concept of symmetric features bring about effective algorithms to precise a number 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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