By Donald S. Passman
First released in 1991, this publication comprises the center fabric for an undergraduate first direction in ring concept. utilizing the underlying subject of projective and injective modules, the writer touches upon numerous facets of commutative and noncommutative ring conception. particularly, a few significant effects are highlighted and proved. half I, 'Projective Modules', starts with simple module thought after which proceeds to surveying a variety of certain sessions of earrings (Wedderbum, Artinian and Noetherian jewelry, hereditary earrings, Dedekind domain names, etc.). This half concludes with an advent and dialogue of the ideas of the projective dimension.Part II, 'Polynomial Rings', reviews those earrings in a mildly noncommutative atmosphere. a number of the effects proved comprise the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for virtually commutative rings). half III, 'Injective Modules', comprises, specifically, quite a few notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian earrings. The booklet includes quite a few workouts and a listing of advised extra examining. it really is compatible for graduate scholars and researchers drawn to ring conception.
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Thus, by the minimality of I again, we have J = 0. But a(e 2 - e) = 0 and e2 - e E J, so it follows that e2 - e = 0.. 0 so e # 0. Thus 0 # eR ~I and we conclude that I= eR. 9 i. ii. iii. iv. If R is a ring, then the following are equivalent. Every R-module is projective. Every R-module is completely reducible. RR is completely reducible. R is a Wedderburn ring. PROOF We first show that (i), (ii), and (iii) are equivalent and then that (iii) is equivalent to (iv). -(ii) If W ~ V are R-modules, then V/W is projective by assumption.
But then e = (ad)k is an idempotent and, by the preceding, Chapter 5. Artinian Rings 49 v(e) = v(ad)k = v(a)k = v(a). Finally, since k ~ 2, we conclude that e = (ad)k = aba for some b ER. D An R-module V is decomposable if it is the direct sum V = Vi -i- V2 of two nonzero submodules. Otherwise, V is indecomposable. The following is the key result on projective modules. 5. 9 Let R be an Artinian ring with nil radical N and suppose Vi, V2, ... , Vm are representatives of the finitely many isomorphism classes of simple R-modules.
Let Ii, 12 , ••• , In be a finite collection of two-sided ideals of R such that Ii+ Ii = R for all i ¥= j. If ai, a2, ... , an are any elements of R, prove that there exists r E R with r ai mod Ii for all i. Deduce that R/(n~ Ii)~ EEl :E~ R/h This is the Chinese Remainder Theorem. 4. Let p be a fixed prime number and let A be the multiplicative group of complex pnth roots of unity for all n ~ 0. If we view A additively as a module over the integers, show that A satisfies min but that it does not have a composition series.
A Course in Ring Theory by Donald S. Passman