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MATH 210 Problem Set 7 Due: March 4, 2016 Exercises 1. Let f : R → S be a homomorphism of commutative rings. (a) For any element r ∈ R and any positive integer n, show that f (nr) = nf (r) and f (rn ) = f (r)n . (b) If I ⊂ R is an ideal of R, and f is surjective, then prove that f (I) is an ideal of S. (c) Show that f is an isomorphism if and only if ker f = h0i and f (R) = S. 2. (a) Which of the following maps from C to C are ring homomorphisms? (i) x + yi 7→ x, (ii) x + yi 7→ x − yi, (iii) x + yi 7→ |x + yi|2 = x2 + y 2 , (iv) x + yi 7→ y + xi. (b) In the ring of integers, find a positive integer n such that hni = h2i + h3i. (c) In the ring of integers, find a positive integer m such that hmi = hai + hbi for any a, b ∈ Z. 3. To try to get comfortable with abstract rings, let’s compare the cases we have seen to the ‘more abstract’ and see how similar they can be. (a) Show that Z/h6i is not a field. (b) Show that (Z/3Z)[x]/hx2 + x + 1i is not a field. (c) Let F be the ring of continuous real-valued functions in one variable, with pointwise addition and multiplication, i.e., for f (x), g(x) ∈ F we have (f + g)(x) = f (x) + g(x) and (f g)(x) = f (x)g(x). Let I be the collection of functions f (x) in F with f (1/3) = f (2/3) = 0. Show that I is an ideal and that F/I is not a field. 1 Problems 4. Let I and J be ideals in a commutative ring R. (a) Prove that IJ ⊂ I ∩ J. (b) If I + J = R, show that I ∩ J = IJ. (c) Give an example of two ideals I, J ⊂ Z such that I ∩ J 6= IJ. 5. Let R be a commutative ring, and let I ⊂ R be an ideal. (a) Show that the nil radical of I, N (I) = {r ∈ R | rn ∈ I for some positive integer n} is an ideal of R. (b) Show that R/N (h0i) has no nonzero nilpotent elements. (N (h0i) is called the nil radical of R.) (c) Suppose I is an ideal in a ring J, and J is an ideal in a ring R. Show that if I contains the multiplicative identity, then I is also an ideal of R. Food for thought A field is a very nice (commutative) ring, it is a ring where all non-zero elements are units, i.e., all but one element is a unit. What about commutative rings where all but two elements are units. Can you classify the possibilities, up to isomorphism? Practice problems (1) Let a R= b 2b a : a, b ∈ R . √ Show that R and Z[ 2] are isomorphic as rings. (See Shifrin 4.1.11.) (2) Let I, J ⊂ R be two ideals in a commutative ring R. Show that I ∪ J is an ideal if and only if I ⊂ J or J ⊂ I. (3) Let I = hx2 + x + 2i ⊂ (Z/5Z)[x]. Find the multiplicative inverse of 2x + 3 + I in (Z/5Z)[x]/I. (4) Shifrin section 4.1: 1, 2, 3, 4, 5, 7, 11, 17. 2