By Donald S. Passman
First released in 1991, this booklet includes the center fabric for an undergraduate first path in ring thought. utilizing the underlying topic of projective and injective modules, the writer touches upon a number of facets of commutative and noncommutative ring concept. particularly, a couple of significant effects are highlighted and proved. half I, 'Projective Modules', starts with easy module concept after which proceeds to surveying numerous certain periods of earrings (Wedderbum, Artinian and Noetherian earrings, hereditary earrings, Dedekind domain names, etc.). This half concludes with an creation and dialogue of the techniques of the projective dimension.Part II, 'Polynomial Rings', reviews those earrings in a mildly noncommutative atmosphere. the various effects proved contain the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for nearly commutative rings). half III, 'Injective Modules', comprises, specifically, a variety of notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian earrings. The booklet includes various routines and an inventory of advised extra studying. it truly is appropriate for graduate scholars and researchers attracted to ring thought.
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Additional info for A Course in Ring Theory
Then Pf PN is a module for the Wedderburn ring RfN and hence it is completely reducible. 7 Uj with U3 9:! 7 Qj, where Q3 9:! ::::! ::::! ::::! 7, P 9:! Q and Pis indeed a direct sum of copies of the Pi 's. Finally, suppose P = · L,kEIC Pk with Pk 9:! Pg(k)· Then we have · LPkfPkN=PfPN= · LU3 kEIC . 5. Thus uniqueness is proved D and the result follows, since each Pi = eiR is a cyclic R-module. In view of the preceding, Pi, P 2 , ••• , Pm are, up to isomorphism, the only nonzero indecomposable ·projective R-modules.
Let K be a field and let X be a set of variables. Then the free Kalgebra K(X) is defined just like the polynomial ring K[X] except that the variables do not commute. In particular, K(X) has a K-basis consisting of all formally distinct words in the variables and it follows easily that this ring has no zero divisors. Prove that R = K (X) has IBN by constructing a homomorphism from R to K. Then show that IXI 2: 2 implies that the regular module RR contains a free submodule of infinite rank. Give a proof, directly from the definition, that a weak direct sum of projective modules is projective.
But V/W is clearly generated by W + Vn and is therefore cyclic. Thus (i) now yields the result. 6(i), completes the proof. D We need one more simple observation. 8 Let I be a right ideal of the ring R. i. I = eR for some idempotent e E R if and only if I is a direct summand of RR· Furthermore, when this happens, we have I= el. Part I. Projective Modules 30 ii. If I is a minimal right ideal, then either 1 2 = 0 or I = eR for some idempotent e. PROOF (i) If I= eR, then R = eR+(l-e)R = I+(l-e)R. Conversely, let R =I+ J and write 1 = e + f withe EI and f E J.
A Course in Ring Theory by Donald S. Passman