Download e-book for kindle: A Course in Ring Theory by Donald S. Passman

By Donald S. Passman

ISBN-10: 0821836803

ISBN-13: 9780821836804

First released in 1991, this ebook comprises the middle fabric for an undergraduate first path in ring thought. utilizing the underlying topic of projective and injective modules, the writer touches upon a variety of features of commutative and noncommutative ring thought. specifically, a few significant effects are highlighted and proved. half I, 'Projective Modules', starts with simple module concept after which proceeds to surveying numerous unique periods of earrings (Wedderbum, Artinian and Noetherian jewelry, hereditary jewelry, Dedekind domain names, etc.). This half concludes with an advent and dialogue of the suggestions of the projective dimension.Part II, 'Polynomial Rings', reports those jewelry in a mildly noncommutative environment. a few of the effects proved contain the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for nearly commutative rings). half III, 'Injective Modules', comprises, particularly, numerous notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian earrings. The publication includes various workouts and a listing of urged extra examining. it's appropriate for graduate scholars and researchers drawn to ring concept.

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Prove that all R-modules are free if and only if R is a division ring. 9. + 4. Wedderburn Rings The goal now is to obtain a more precise description of Wedderburn rings. We do this by computing the endomorphism ring of certain module direct sums. orphisms on the left. The following lemma is, for the most part, fairly obvious. Let V and W be R-modules. i. HomR(V, W) is a right EndR(V)-module and a left EndR(W)module with multiplication given by function composition. ii. If V = · E~=l Vi is a finite direct sum, then HomR(V, W) is isomorphic to$ E~=l HomR(Vi, W) as a left EndR(W)-module.

6. Let R = Mn(D) with D a division ring. Show that R has finitely many right ideals if and only if either n = 1 or D is finite. 7. Let K be a field and let R ~ M2(K) be given by R = ( ~ ~). If e = e1 , 1 , show that EndR(eR) is a field even though eR is not an irreducible R-module. 8. Write R = ·E~= 1 Mn,,(Dk) as in the Artin-Wedderburn Theorem. If I

Furthermore, the map O! H a 1 $ a2 $···$an is th,en easily seen to be an EndR(W)-module isomorphism from HomR(V, W) to $ Ei HomR(Vi, W). 33 Part I. Projective Modules 34 (iii) Each element {3 e HomR(V, W) determines R-homomorphisms {33: V ~ W ~ W3 for j = 1, 2, ... , m via composition. It is then easy to see that the map {3 1-+ {31 $ {32 $ · · · $ f3m is an EndR(V)-module isomorphism from HomR(V, W) to$ Et HomR(V, W3). (iv) Here the result follows as in (ih), provided that each such {3 has at most finitely many nonzero {33.

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