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?

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

Yes, this works perfectly fine. A ring is in particular a group with respect to the addition operation, and any ideal is in particular a subgroup. So any general statement about groups and subgroups will also be true for rings and ideals.

(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.)

  • $\begingroup$ What if $R$ and $I$ were finite rings such that $|I|$ divides $|R|$ $\endgroup$ – user3000482 Dec 18 '16 at 23:58
  • $\begingroup$ Sorry, I don't understand what you're trying to ask. $\endgroup$ – Eric Wofsey Dec 19 '16 at 0:05
  • $\begingroup$ @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 $|R|$? $\endgroup$ – D_S Dec 19 '16 at 0:43
  • $\begingroup$ @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 $R$? $\endgroup$ – user3000482 Dec 19 '16 at 0:57
  • $\begingroup$ @D_S If $I$ was an ideal of $R$, wouldn't $|I|$ divide $|R|$ if we use Lagrange's theorem? $\endgroup$ – user3000482 Dec 19 '16 at 0:58

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