Download e-book for kindle: a course in ring theory by Donald S. Passman

By Donald S. Passman

First released in 1991, this booklet comprises the middle fabric for an undergraduate first path in ring thought. utilizing the underlying subject of projective and injective modules, the writer touches upon a variety of facets of commutative and noncommutative ring thought. specifically, a few significant effects are highlighted and proved. half I, 'Projective Modules', starts with uncomplicated module thought after which proceeds to surveying a number of distinct sessions of earrings (Wedderbum, Artinian and Noetherian jewelry, hereditary earrings, Dedekind domain names, etc.). This half concludes with an advent and dialogue of the recommendations of the projective dimension.Part II, 'Polynomial Rings', stories those earrings in a mildly noncommutative environment. a few 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', contains, particularly, a number of notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian earrings. The publication comprises a variety of routines and a listing of prompt extra examining. it's appropriate for graduate scholars and researchers drawn to ring thought.

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We next need the equivalent of the cancellation property (Lemma 1). The original proof is invalid since ZM is not necessarily an integral domain (see remarks at the end of this section). Lemma 7 The cancellation property holds: 2K−1 j=0 0(mod M ) if s ≡ 0 mod 2K, 2K(mod M ) if s ≡ 0 mod 2K. ω js ≡ Proof. The result is true if s ≡ 0 mod 2K. Assuming otherwise, let (s mod 2K) = 2p q where q is odd, 0 < 2p < 2K and let r = 2K · 2−p > 1. Then by breaking up the desired sum into 2p parts, ω js ω js + = j=r j=0 j=0 2K−1 2r−1 r−1 2K−1 ω js + · · · + ω js j=2K−r r−1 r−1 r−1 p ω js + ω rs = j=0 j=0 ω js + · · · + ω rs(2 ω js −1) j=0 r−1 2p ≡ since ω rs j=0 ≡ 1 mod M .

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