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 examine degrees. the most necessities are familiarity with teams, jewelry and fields. Proofs are self-contained.

The booklet could be helpful to newcomers and skilled researchers alike. the fabric is so prepared that the newbie can study via self-study or by means of attending a direction. For the skilled researcher, the ebook may possibly serve to offer new views on a few famous effects, or as a reference.

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Since A ⊆ A[X1 ] ⊆ · · · ⊆ A[X1 , . . , Xn ] ⊆ A[X1 , . . , Xn , Xn+1 ] ⊆ · · · , the union of this sequence of commutative A-algebras is a commutative Aalgebra. This union is denoted by A[X1 , X2 , X3 , . ] and is defined to be the polynomial ring over A in the countably many variables X1 , X2 , X3 , . . Note that any given element of this ring is a polynomial in finitely many of the variables. 1 Substitution and Universal Property. Let B be a commutative Aalgebra, and let b1 , . . , bn be n given elements of B.

Clear. We say that N is the internal direct sum of the family of submodules {Ni }i∈I , and we write N = i∈I Ni , if any of the equivalent conditions of the above proposition holds. 1. 1 that i∈I Mi is the internal direct sum of the family {qi (Mi )}i∈I . In view of this, we usually identify direct sum and internal direct sum in a natural way, and speak only of the direct sum. 4. 4 23 Free Modules In this section, we assume that A is a nonzero ring. Let M be an A-module. A system S = {si }i∈I of elements of M is said to be linearly independent (over A) if the condition i∈I ai si = 0 with ai ∈ A for every i and ai = 0 for almost i implies that ai = 0 for every i.

Minimal among all prime ideals of A) is homogeneous. (5) A is a field if and only if A = A0 and A0 is a field. (6) If m0 is a maximal ideal of A0 then m0 + A+ is a homogeneous maximal ideal of A. Further, every homogeneous maximal ideal of A is of the form m0 + A+ with m0 a maximal ideal of A0 . (7) Every proper homogeneous ideal of A is contained in a homogeneous maximal ideal of A. Proof. (1) Let a and b be nonzero elements of A. We have to show that ab = 0. For each n, let an and bn be the homogeneous components of a and b, respectively, of degree n.