18. Let (R, 1.) be a commutative ring with identity and let N denote the set of nilpotent elements of R. Verify that a) the triple (N, +,) is an ideal of (R, +,). (llint: If a - 6m - 0, consider (a –...

Extracted text: 18. Let (R, 1.) be a commutative ring with identity and let N denote the set of nilpotent elements of R. Verify that a) the triple (N, +,) is an ideal of (R, +,). (llint: If a - 6m - 0, consider (a – b)*+m) b) the quotient ring (R/N, +,) has no nonzero nilpotent elements. 19. Assume (R, +,) is a ring with the property that a? + a € cent R for every element a in R. Show that (R,+,) is a commutative ring. [Hint: Make use of the expression (a + b)2 + (a + b) to prove, first, that a-b+b.a lies in the center.) 20. Illustrate Theorem 3-18 by considering the rings (Zo, +o, a), (Z3, +3, 3), and the homomorphism f: Zo 2 defined by s10) = (3) - 0, (1) = (4) = 1, s(2) = f(5) %3D 2.