Get A Course in Ring Theory PDF

By Donald S. Passman

ISBN-10: 0821836803

ISBN-13: 9780821836804

First released in 1991, this booklet comprises the center fabric for an undergraduate first direction in ring conception. utilizing the underlying subject of projective and injective modules, the writer touches upon a number of elements of commutative and noncommutative ring conception. specifically, a couple of significant effects are highlighted and proved. half I, 'Projective Modules', starts off with uncomplicated module conception after which proceeds to surveying a number of certain periods of jewelry (Wedderbum, Artinian and Noetherian jewelry, hereditary jewelry, Dedekind domain names, etc.). This half concludes with an creation and dialogue of the techniques of the projective dimension.Part II, 'Polynomial Rings', experiences those jewelry in a mildly noncommutative surroundings. many of the effects proved contain the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for virtually commutative rings). half III, 'Injective Modules', contains, particularly, quite a few notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian jewelry. The booklet includes various workouts and an inventory of urged extra interpreting. it's compatible for graduate scholars and researchers drawn to ring conception.

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L(iii), Nil(R) is, in fact, the unique largest nil two-sided ideal of R. Furthermore, by (ii) we see that Nil(R/Nil(R)) = 0. DEFINITION Let I <1 R. i. If W is an R/ I-module, then W is naturally an R-module with multiplication given by wr = w(r+l) for all w E W, r ER. module homomorphism W-+ W' is also an R-homomorphism. ii. module with multiplication given by v(r + J) = vr for all v E V, r E R. Furthermore, the R/ I-submodules of V are precisely the R-submodules and any R-homomorphism V-+ V' is also an Rf I-homomorphism.

AnnR(W) = {r E R I Wr = 0 }, is a right ideal of R. We first show that the double annihilator condition holds for finite dimensional subspaces of V. annR(W) = O, then x E W. We proceed by induction on dimD = m < oo, the case m = 0 being trivial. Thus suppose that m ~ 1 and write W = W' Dw, where W' is a subspace of dimension m - 1. annR(W'), then wA' :/:. O. But A' is a right ideal of R, so wA' is a nonzero R-submodule of V and hence wA' = V. w + Part I. aDllR(W) and let x E V with xA = 0. The goal is to show that x E W.

8, and obviously R must act faithfully on V. Thus any simple ring is necessarily primitive. DEFINITION Chapter 4. Wedderburn Rings 39 Now let R act faithfully on the irreducible module V and set D = EndR(V). Then Dis a division ring, by Schur's Lemma, and as we have observed, V is a left EndR(V)-module. Thus V is a left D-vector space. Furthermore, if d ED, v EV and r ER, then the associativity condition d(vr) = (dv)r implies that r is a D-endomorphism of V or, equivalently, a D-linear transformation.

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A Course in Ring Theory by Donald S. Passman

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