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freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Algebra Help

I have this question that I'm slightly stuck on:
Let R be a ring with identity. We say that two elements a and b of R are associates,
written a ~ b, if b = au for some unit u of R.
In the case that R = Z12, calculate the equivalence classes of ~

How do I go about answering this?
Thanks in advance

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Algebra Help

Well, {0} is obviously an equivalence class unto itself.

In the integers modulo 12, there four units, namely 1, 5, 7, 11 (these are the integers that are coprime with 12).

Thus U = {1,5,7,11} (the set of units) is another equivalence class.

To find the other equivalence classes, just “multiply” each of the other nonzero elements with U.

2U = {2,10} = 10U
3U = {3,9} = 9U
4U = {4,8} = 8U
6U = {6}

Hence the equivalence classes are {0}, {1,5,7,11}, {2,10}, {3,9}, {4,8}, {6}.

Last edited by JaneFairfax (2007-03-14 03:33:59)

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Re: Algebra Help

Appreciate that thanks!