Exercise 3: Let R be a commutative ring. (a) Let I and J be ideals in R. Show that In J is also an ideal in R. (b) Now let R be a principal ideal domain. Let a, b E R and let (a) (b) = (m) for some me R. Show that m is the least common multiple of a and b, written m = lcm(a, b), meaning that a | m, bm, and if c is an element such that ac and b | c, then m | c. [Note that m is defined only up to multiplication by a unit, by Exercise 1.] (c) Still assuming that R is a PID, suppose that (a, b) = (1) = R. Show that (a) n (b) = (ab). (Hint: You can write 1 = au + bv for some u, v € R.)

Let S = {a + bi | a, b e Z, b is even}. Show that S is a subring of Z[i], but not an ideal of Z[i].

Consider the proof below of the fact that I + a = 1 + b iff a - bEI where R is a commutative right with unity and I is an ideal. Identify which “+" signs are notation and which "+" signs are operations. (=) Assume I + a = 1 + b. : a E l + a : a el + b . a = i+ b for some i EI : i = a – b i a – b E l (=) Assume a – b E I Assume ΧΕ α :x = i + a for some i E I :x = i+ a + (-b+ b) :x = (i + a - b) + b i x E I + b : I + a CI+ b Assume x E I+ b :x = i + b for some i El :x = i +b+ (-a + b) : x = (i + b – a) + a : x € l + a :I+b SI+a : I+ a = 1 + b Consider the proof above and justify each claim.

Consider the ring R = 2Z and the ideal I = 24Z of R. Give a representative for each coset of I in R.