Math 515 homework #3 Due date: 19980128 D. J. Bernstein 19980121

Ideals. Let R be a ring. A subset I ⊆ R is an ideal of R if 0 ∈ I; −a ∈ I whenever a ∈ I; a + b ∈ I whenever a ∈ I and b ∈ I; and ra ∈ I whenever r ∈ R and a ∈ I.

Quotients. In the following exercises, I is an ideal of a ring R. The quotient R/I is the collection {r + I : r ∈ R}, where r + I means {r + a : a ∈ I}. There is a natural map r ↦ r + I from R to R/I. The purpose of these exercises is to define a ring structure on R/I making the natural map R → R/I a ring homomorphism. For example, the set Z/3Z = {0 + 3Z, 1 + 3Z, 2 + 3Z}, abbreviated Z/3, is a ring; the natural map Z → Z/3Z takes 38 to 38 + 3Z = 2 + 3Z. Exercise 6. Consider the map m : R × R → R/I defined by m(a, b) = ab + I. Prove that m induces a binary operation on R/I: there is a map φ : R/I × R/I → R/I such that φ(a + I, b + I) = ab + I. Similar comments apply to addition and negation. The induced maps, together with 0 + I and 1 + I, form a (0, 1, −, +, ·) structure on R/I. Exercise 7. Prove that r ↦ r + I is a (0, 1, −, +, ·) homomorphism from R to R/I. Exercise 8. Prove that R/I is a ring. (Hint: The homomorphism R → R/I is surjective; use it to push identities from R to R/I.)