Basic Commutative Algebra by Balwant Singh

By Balwant Singh

This textbook, set for a one or semester direction in commutative algebra, presents an creation to commutative algebra on the postgraduate and study degrees. the most must haves are familiarity with teams, earrings and fields. Proofs are self-contained.

The e-book might be necessary to novices and skilled researchers alike. the cloth is so prepared that the newbie can study via self-study or through attending a path. For the skilled researcher, the booklet may well serve to give new views on a few famous effects, or as a reference.

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To show existence, let S be a given set. Let F = s∈S As , where each As = A. For s ∈ S, let qs : As → F be the canonical inclusion, and let es = qs (1). Let T = {es | s ∈ S}. Then, noting that qs (as ) = as es , it follows that F is free with basis T. 2, F is free on T. Now, since the map S → T given by s → es is clearly bijective, F is free on S. In view of the uniqueness, we may call a free A-module (F, j) on S the free A-module on S. 2, we regard S as a subset of F . Then F is free with basis S.

3. 3 Uniqueness. In general, if an object defined via a universal property exists then it is easy to see that it is unique up to a unique isomorphism. Here, the uniqueness of the isomorphism requires, of course, that the isomorphism be compatible with the given data. Let us illustrate this by giving an argument for the uniqueness of the categorical direct sum defined above. Suppose (S, {µi }i∈I ) and (S , {µi }i∈I ) are two categorical direct sums of the family {Mi }i∈I . By the universal property of the first pair, there exists a unique A-homomorphism ψ : S → S such that µi = ψµi for every i.

This module is called the direct product of the family {Mi }i∈I . Modules and Algebras 20 The A-submodule i∈I of i∈I Mi := {(xi ) ∈ i∈I Mi | xi = 0 for almost all i} Mi is called the direct sum of the family {Mi }i∈I . The direct product and direct sum of a finite family of modules M1 , M2 , . . , Mn are also written, respectively, as M1 × M2 × · · · × Mn and M1 ⊕ M2 ⊕ · · · ⊕ Mn . If the indexing set I is finite then the direct product and the direct sum coincide, so either terminology or notation can be used.

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