1.

a.Consider the groups R^ = (R{0}, ) and R^+ = ((0, ), ). Find a homomorphism : R {0} (0, ) which has kernel 1. Prove that your has the desired properties. What does the fundamental homomorphism theorem tell us about the group R/1?

b. An element x in a commutative ring is called idempotent if x^2 = x. Prove that 0 and 1 are the only two idempotents in an integral domain. Then give an example of a commutative ring with an idempotent that is not 0 or 1.