Topics covered: preliminaries from commutative algebra. Note: throughout, by a commutative ring we mean a commutative associative unital ring. A commutative ring homomorphism: A B is a map that is a homomorphism (4.+) → (B.+) of the additive groups of A and B. preserves the product p(ry)=(z)(y) and maps the identity element of A to that of B. 1.1. Let A be a commutative ring and a CA be an ideal of A. Prove that there is a one-to-one correspondence (that preserves the order by inclusion) between the set of ideals b of A containing a and the set of all ideals of the quotient ring A/a. 1.2. (Ideals generated by sets) Let S be a subset of a commutative ring. Denote by (S) (also common is the notation (S)) the set of all finite A-linear combinations of elements of S: 1+₂++. Ti € S, a, € A. A. Verify that (S) is an ideal of A. B. Prove that (S) is the smallest ideal of A containing the subset S (and in fact (S) coincides with the intersection sgpb of all ideals b of A that contain S).
Topics covered: preliminaries from commutative algebra. Note: throughout, by a commutative ring we mean a commutative associative unital ring. A commutative ring homomorphism: A B is a map that is a homomorphism (4.+) → (B.+) of the additive groups of A and B. preserves the product p(ry)=(z)(y) and maps the identity element of A to that of B. 1.1. Let A be a commutative ring and a CA be an ideal of A. Prove that there is a one-to-one correspondence (that preserves the order by inclusion) between the set of ideals b of A containing a and the set of all ideals of the quotient ring A/a. 1.2. (Ideals generated by sets) Let S be a subset of a commutative ring. Denote by (S) (also common is the notation (S)) the set of all finite A-linear combinations of elements of S: 1+₂++. Ti € S, a, € A. A. Verify that (S) is an ideal of A. B. Prove that (S) is the smallest ideal of A containing the subset S (and in fact (S) coincides with the intersection sgpb of all ideals b of A that contain S).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Transcribed Image Text:Topics covered: preliminaries from commutative algebra.
Note: throughout, by a commutative ring we mean a commutative associative unital ring. A
commutative ring homomorphism: A B is a map that is a homomorphism (4.+) → (B.+)
of the additive groups of A and B. preserves the product p(ry)=(z)(y) and maps the identity
element of A to that of B.
1.1. Let A be a commutative ring and a CA be an ideal of A. Prove that there is a one-to-one
correspondence (that preserves the order by inclusion) between the set of ideals b of A containing
a and the set of all ideals of the quotient ring A/a.
1.2. (Ideals generated by sets) Let S be a subset of a commutative ring. Denote by (S) (also
common is the notation (S)) the set of all finite A-linear combinations of elements of S:
1+₂++. Ti € S, a, € A.
A. Verify that (S) is an ideal of A.
B. Prove that (S) is the smallest ideal of A containing the subset S (and in fact (S) coincides
with the intersection sgpb of all ideals b of A that contain S).
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