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

Section: Chapter Questions

Problem 1RQ

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