Show that the subset S 0 4 8 12 16 20 of Z24 is a subring D

Show that the subset S = {[0], [4], [8], [12], [16], [20]} of Z24 is a subring. Does S have an identity?

Solution

Note that we can characterize the set S as S = {[4q]24 : q Z}. By inspection we see that [0]24 S. Also, for any a, b S we have a = [4q]24 and b = [4r]24 for some integers q, r Z. So a + b = [4q] + [4r] = [4(q + r)] S, ab = [4q][4r] = [4(4qr)] S, and a = [4q] = [4(q)] S. By Theorem 3.2, S is a subring of Z24

Hence Proved.

Yes S have an identity.