Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 10 November 10, 2005

Problem 1: Let R be a commutative ring with 1 ≠ 0. Let J be the nilradical of R. This is the the ideal consisting of the nilpotent elements of R, i.e., the set of elements a ∈ R such that a^{n} = 0 for some nonnegative integer n. (Cf. Problem 1 on sheet 8.)

(a) Prove that J is contained in the intersection of all maximal ideals I of R:

J ⊂ ⋂_{I maximal} I.

(b) Prove that J equals the intersection of all prime ideals I of R:

J = ⋂_{I prime} I.

(Hint: Zorn’s Lemma).

Problem 2: Let R be an integral domain and assume that every prime ideal in R is principal. Prove that every ideal of R is principal, i.e., R is a principal ideal domain. (Hint: Zorn’s Lemma).

Problem 3: Let R be a unique factorization domain. Assume that for any pair of elements a, b ∈ R, the ideal I = (a, b) they generate is a principal ideal. Prove that R is a principal ideal domain.

Problem 4: Let R be a principal ideal domain.

(a) Let a_{1}, …, a_{n} ∈ R. Prove that for a greatest common divisor d of a_{1}, …, a_{n} there exist elements x_{1}, …, x_{n} ∈ R such that

d = x_{1}a_{1} + ⋯ + x_{n}a_{n}.

(b) Let a, b, c ∈ R. Show that if a and b are relative prime that b|ac implies b|c.