Math 514 homework #2 Due date: 19950901 D. J. Bernstein 19950824 Let R be a ring, and let I be a subset of R. We call I an ideal of R if I satisfies three conditions: (1) 0 ∈ I;

(2) x − y ∈ I if x ∈ I and y ∈ I; (3) rx ∈ I if r ∈ R and x ∈ I. Exercise 4. Let E be any subset of a ring R. Let I be the intersection of all ideals of R that contain E. Prove that I is an ideal of R. (Hence I is the smallest ideal of R containing E.) Let D be a domain. An element p ∈ D is called prime if (1) p ≠ 0; (2) p is not a unit; (3) if p divides ab then p divides a or b. For example, the prime elements of Z are {…, −7, −5, −3, −2, 2, 3, 5, 7, …}. The positive prime elements of Z are called prime numbers; no such terminology applies to rings other than Z. Exercise 5. Prove that if p is a prime and u is a unit then up is a prime. Exercise 6. Prove that if p divides a product b_{1}b_{2} … b_{s} then p divides b_{i} for some i. (Hint: Induct on s. Don’t forget s = 0.) A unique factorization domain is a domain in which every nonzero element is a unit or a product of prime elements. For example, Z is a unique factorization domain. Exercise 7. Let D be a unique factorization domain. Prove that factorizations in D are, in fact, unique (up to units): if p_{1}p_{2} … p_{r} = q_{1}q_{2} … q_{s}, then (1) r = s; (2) there is a permutation π : {1, 2, …, r} → {1, 2, …, r}, and units u_{1}, u_{2}, …, u_{r}, such that p_{i} = u_{i}q_{π(i)}. (Hint: Induct on r.)