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By B. Osofsky

Those notes have been ready for a sequence of ten lectures given at local convention of the convention Board of the Mathematical Sciences in June 1971. the majority of the lectures have been on projective dimensions of ``very large'' modules. even though the cloth in those notes isn't really new, there are numerous areas the place present paintings has been simplified. for instance, a commutative neighborhood nondomain of world size three is defined regardless of research, and the size of a quotient box of a polynomial ring instead of a standard neighborhood ring is calculated. A derivation of Tor, one step at a time with no the standard derived functor equipment, can also be incorporated.

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The most goal of those lectures is first to in brief survey the elemental con­ nection among the illustration concept of the symmetric workforce Sn and the idea of symmetric capabilities and moment to teach how combinatorial equipment that come up evidently within the thought of symmetric services result in effective algorithms to specific quite a few 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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