Definition. Let be a ring and let be a two-sided ideal (http://planetmath.org/Ideal) of . To define the quotient ring , let us first define an equivalence relation in . We say that the elements are equivalent, written as , if and only if . If is an element of , we denote the corresponding equivalence class by . Thus if and only if . The quotient ring of modulo is the set , with a ring structure defined as follows. If are equivalence classes in , then
Here and are some elements in that represent and . By construction, every element in has such a representative in . Moreover, since is closed under addition and multiplication, one can verify that the ring structure in is well defined.
For a ring , we have and .
|Date of creation||2013-03-22 11:52:32|
|Last modified on||2013-03-22 11:52:32|
|Last modified by||mathwizard (128)|