The ideal quotient 
,
also called a colon ideal, is the ideal in a commutative
ring 
defined by
 |
where 
and 
are ideals. It is an analog of division
for ideals.
If 
, then 
iff 
for every 
, ..., 
. In particular, quotienting by a principal
ideal 
requires only 
(Jensen 2014, p. 67).
For positive integers 
and 
, the ring of integers gives
 |
where 
is the greatest common divisor. Thus 
, while
. Taking
the union of the quotients by successive powers
of an ideal gives ideal
saturation.
