Notice that the restricted operations inherit the associative and distributive properties of and , as well as commutativity of . So for to be a ring by itself, we need that be a subgroup of and that be closed. The subgroup condition is equivalent to being non-empty and having the property that for all .
A subring is called a left ideal if for all and all we have . Right ideals are defined similarly, with instead of . If is both a left ideal and a right ideal, then it is called a two-sided ideal. If is commutative, then all three definitions coincide. In ring theory, ideals are far more important than subrings, as they play a role analogous to normal subgroups in group theory.
|Date of creation||2013-03-22 12:30:19|
|Last modified on||2013-03-22 12:30:19|
|Last modified by||yark (2760)|