Quotient ring
A quotient ring is a quotient set of the elements of a ring with an induced ring structure.
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
.