Ring theory- zero divisors and integral domains

[porroadventum]

Homework Statement

Consider the ring Z/mZ, show that S = {[0], [a], [2a], · · · , [m − a]} forms a (possibly
nonunitary) subring of Z/mZ when a divides m. (i.e. show that (S,+, ·) is closed
the usual addition and multiplication. (We are not require to find a multiplicative identity).

The Attempt at a Solution

Since a divides m then m=ab so I tried subbing in ab for m and got [m-a]=[ab-a]=[a(1-b)]... but not too sure where to go from here. From looking at the set S it does not seem to be closed under addition or multiplication? Just a hint at how to go about/start/ approach this question would much appreciated! THank you!

 

[jbunniii]

Well, let's check closure under addition. Two general elements of $S$ look like $[ra]$ and $[sa]$, where $r$ and $s$ are integers. So what is $[ra] + [sa]$? Evaluate it in the ring $\mathbb{Z}/m\mathbb{Z}$, and see if the answer is in $S$.

 

[porroadventum]

OK, I see how to prove it now- the addition of two elements of S will always give a number which is a multiple of a, therefore will be an element of S. (Similarly for multiplication)...?

(Thank you)

 

[jbunniii]

Yes, similarly for multiplication.