radical of an ideal
The radical of an ideal is always an ideal of .
If , then is called a radical ideal.
More generally, the radical of an ideal in can be defined over an arbitrary ring. Let be an ideal of a ring , the radical of is the set of such that every m-system containing has a non-empty intersection with :
Under this definition, we see that is again an ideal (two-sided) and it is a subset of . Furthermore, if is commutative, the two sets coincide. In other words, this definition of a radical of an ideal is indeed a “generalization” of the radical of an ideal in a commutative ring.
|Title||radical of an ideal|
|Date of creation||2013-03-22 12:35:54|
|Last modified on||2013-03-22 12:35:54|
|Last modified by||CWoo (3771)|