# Can I use Lagrange's Theorem for rings with ideals?

I know we can use Lagrange's Theorem for groups, but can we also use them for rings? So, suppose we have $R$ as a ring and $I$ as an ideal. Then would it be true to say $R$ only has $|R|/|I|$ distinct cosets? What are the limitations?

• At least for finite rings maybe? – user3000482 Dec 18 '16 at 23:52

(Note that it doesn't really make sense to say there are $|R|/|I|$ distinct cosets when these sets could be infinite, division of infinite cardinalities cannot always be uniquely defined. But what is true always is that $|R|=|I|\cdot|R/I|$. This doesn't have anything to do with rings in particular though; this is just the correct way to state Lagrange's theorem for infinite groups.)
• What if $R$ and $I$ were finite rings such that $|I|$ divides $|R|$ – user3000482 Dec 18 '16 at 23:58
• @OP do you want $I$ and $R$ to have some relationship with one another? Or do you just want them to be some arbitrary finite rings where $|I|$ divides $|R|$? – D_S Dec 19 '16 at 0:43
• @EricWofsey I was trying to ask, if $R$ and $I$ were finite, then $|R|/|I|$ will be the number of distinct cosets of $I$ in $R$? – user3000482 Dec 19 '16 at 0:57
• @D_S If $I$ was an ideal of $R$, wouldn't $|I|$ divide $|R|$ if we use Lagrange's theorem? – user3000482 Dec 19 '16 at 0:58