1. Consider the proof of the backwards direction of the theorem below. Justify each step that makes a claim, no matter how trivial. It is available in a Word document for your convenience. *** Let R be a commutative ring with unity and I an ideal of R. The quotient ring R/I is an integral domain if and only if I is prime.** (=) Assume I is a prime ideal of R. Assume a, b ER/I such that ab = .. a = 1 + a2 : b = 1 + b2 for some b2 E R. :: (1 + az)(I + b2) = 1 + OR :I + azb, = 1 + OR : azb, – OR E I . azbz E I : az El or b, € I Wlog az E I = OR/I- for some a, ER. az – OR E I :I+ az = I +0R Or/I : R/l is an integral domain. :: a =

Exercise 44. Show that 1 and -1 are units in any ring R. Show that if R is a ring with 0 # 1 then 0 is not a unit. (Most rings have 0 + 1. The trivial ring is an exception: it has only one element so all elements are equal).