By Donald S. Passman
First released in 1991, this ebook comprises the middle fabric for an undergraduate first direction in ring thought. utilizing the underlying subject matter of projective and injective modules, the writer touches upon numerous features of commutative and noncommutative ring concept. particularly, a couple of significant effects are highlighted and proved. half I, 'Projective Modules', starts off with easy module conception after which proceeds to surveying numerous unique periods of jewelry (Wedderbum, Artinian and Noetherian earrings, hereditary jewelry, Dedekind domain names, etc.). This half concludes with an creation and dialogue of the suggestions of the projective dimension.Part II, 'Polynomial Rings', reports those jewelry in a mildly noncommutative environment. many of the effects proved comprise the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for nearly commutative rings). half III, 'Injective Modules', contains, specifically, numerous notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian earrings. The ebook includes quite a few routines and a listing of urged extra analyzing. it truly is appropriate for graduate scholars and researchers drawn to ring thought.
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Additional resources for A Course in Ring Theory
Then since V is generated by all Vi, it follows that there exists some Vk with Vk Sf; W'. But Vk is irreducible, so this implies that W' n Vk = 0 and hence that W' + Vk is a direct sum. 1' Vj) is also direct and hence J' ES. J and thus W' = V as required. D +(• + It is now a simple matter to characterize completely reducible modules. For an R-module V, the following are equivalent. i. V is a sum of irreducible submodules. ii. V is a direct sum of irreducible submodules. iii. V is completely reducible.
1' Vj) is also direct and hence J' ES. J and thus W' = V as required. D +(• + It is now a simple matter to characterize completely reducible modules. For an R-module V, the following are equivalent. i. V is a sum of irreducible submodules. ii. V is a direct sum of irreducible submodules. iii. V is completely reducible. -(i). To this end, let V be completely reducible and let S be the sum of all simple submodules of V. The goal is to show that S = V. If this is not the case, fix v EV\ S. By Zorn's Lemma we can choose a submodule M of V maximal with the properties 8 ~ M and v fj.
Suppose first that fJ( v) = 0. Then 0 = fJ(v)e = ve = v, since v E Ve, and hence() is one-to-one. On the other hand, if a E HomR(eR, V), set w = a(e). Then w = a(e·e) = a(e)e =we, Part I. Projective Modules 36 sow= we E Ve. But for any r ER, we have a(er) = a(e)r = (we)r = w(er), so a= O(w) and e is onto. (ii) Here we need only observe that if V = eR, then e preserves multiplication. For this, let s, t E eRe and let r E eR. Then O(s)O(t)r = e(s)tr = str = e(st)r and this fact is proved. (iii) It follows from statement (ii), with e = 1, that EndR(R) ~ R.
A Course in Ring Theory by Donald S. Passman