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Download e-book for iPad: Basic Commutative Algebra by Balwant Singh

By Balwant Singh

ISBN-10: 9814313610

ISBN-13: 9789814313612

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

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

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Given f, g ∈ HomA (M, N ) and a ∈ A, define maps f + g : M → N and af : M → N by (f + g)(x) = f (x) + g(x) and (af )(x) = a(f (x)) for x ∈ M. Then these maps belong to HomA (M, N ), and it is easy to see that these operations make HomA (M, N ) an A-module. On defining multiplication in HomA (M, M ) as composition of maps, HomA (M, M ) becomes a ring (usually noncommutative) with 1M as the multiplicative identity. If A is not necessarily commutative then HomA (M, N ) is an additive group and HomA (M, M ) is a ring under the operations defined above but these are not A-modules in general because, in the above notation, af need not be an A-homomorphism.

We have f = f + aX r with deg f < r, and 0 = f g = f g. Therefore, by induction hypothesis, all coefficients of f are zerodivisors in A. Also, a is a zerodivisor in A because ab = 0. Thus all coefficients of f are zerodivisors in A. 2 Power Series Rings In this section, we find it convenient to work with t (rather than n) indeterminates. Let A be a ring and let X1 , . . , Xt be indeterminates. The power series ring in the variables X1 , . . , Xt over A, denoted A[[X1 , . . , Xt ]], consists of all (formal) power series in X1 , .

We have f = f + aX r with deg f < r, and 0 = f g = f g. Therefore, by induction hypothesis, all coefficients of f are zerodivisors in A. Also, a is a zerodivisor in A because ab = 0. Thus all coefficients of f are zerodivisors in A. 2 Power Series Rings In this section, we find it convenient to work with t (rather than n) indeterminates. Let A be a ring and let X1 , . . , Xt be indeterminates. The power series ring in the variables X1 , . . , Xt over A, denoted A[[X1 , . . , Xt ]], consists of all (formal) power series in X1 , .

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Basic Commutative Algebra by Balwant Singh


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