subring
Let a ring. A subring is a subset of with the operations and of restricted to and such that is a ring by itself.
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.
Example:
Consider the ring ). Then is a subring, since the difference and product
of two even numbers is again an even number.
Title | subring |
Canonical name | Subring |
Date of creation | 2013-03-22 12:30:19 |
Last modified on | 2013-03-22 12:30:19 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 17 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 20-00 |
Classification | msc 16-00 |
Classification | msc 13-00 |
Related topic | Ideal |
Related topic | Ring |
Related topic | Group |
Related topic | Subgroup |
Defines | ideal |