Characterization of Equivalence Relations Compatible with Ring Structure
Theorem. Let be an equivalence relation on the underlying set of a pseudo-ring . Then is compatible with addition and left (resp. right) multiplication if and only if is equivalent to a statement of the form "", for some left (resp. right) ideal of .
Proof. We prove the case for left ideals; the other case follows from passing to the opposite ring.
Suppose is an equivalence relation on compatible with addition and left multiplication. Let be the equivalence class of 0. Then is evidently equivalent to the statement ", so it remains to show that is a left ideal of .
By definition, , and for any , so ; that is, is closed under addition. Finally, for any and , so . Therefore is a left ideal of .
Conversely, let be any left ideal of . We wish to show that "" is an equivalence relation compatible addition and left multiplication in . Evidently, if and , then so . Also, is an element of , and if is, then so is . This shows that equivalence modulo is an equivalence relation.
Now we show that equivalence modulo is compatible with addition and left multiplication. Indeed, suppose that ; then for any , so . Finally, for any , since is a left ideal of .
Corollary. Let be a ring, and an equivalence relation on the elements of . Then is compatible with the ring structure of if and only if it is of the form "", for some two-sided ideal of .