Subring Example

Question

Let S={[0],[1],[2],[3],[4]}. Is S a subring of Z(mod 6)?

My thoughts: I have a lot of trouble sort of working with the sets I am given in certain situations like these. I know the first two qualifications of a subring are closed under addition and multiplication. So do I just take the individual congruence classes [0] and [2] for example and add combinations together to see if their addition concludes that it isint a subring?

Answer 1

$|S|=5$ and $|\Bbb Z_6|=6$, so isn't even a subgroup, since $5$ doesn't divide $6$..

Answer 2

Yes, you are right. If you want to do it 'by foot' you could say for example that $[1]+[4]=[5]$ is not in the set, and you are done. Obviously you can answer the question using an abstract argument as well (see @ajotatxe's answer).