Verify that I 0 3 is an ideal in zopf6 and list all its dis

Verify that I = {0, 3} is an ideal in zopf_6 and list all its distinct cosets. Verify that I = {0, 3, 6, 9, 12}is an ideal in zopf_15 and list all its distinct cosets.

Solution

Addition:
0 + 0 = 0, 3 + 3 = 0, and 0 + 3 = 0

Multiplication:
n * 0 = 0 for any n in Z6, and n * 3 = 0 if n is even (mod 6) and 3 otherwise.
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Since |I| = 2 and |Z6| = 6, there will be |Z6|/|I| = 6/2 = 3 distinct cosets.
They are 0 + I = {0, 3}, 1 + I = {1, 4}, and 2 + I = {2, 5}.
(Note that their union is Z6.