Relation To Nilpotent Ideals
The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil. There are two main barriers for nil ideals to be nilpotent:
- There need not be an upper bound on the exponent required to annihilate elements. Arbitrarily high exponents may be required.
- The product of n nilpotent elements may be nonzero for arbitrarily high n.
Clearly both of these barriers must be avoided for a nil ideal to qualify as nilpotent.
In a right artinian ring, any nil ideal is nilpotent. This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the artinian hypothesis), the result follows. In fact, this has been generalized to right noetherian rings; the result is known as Levitzky's theorem. A particularly simple proof due to Utumi can be found in (Herstein 1968, Theorem 1.4.5, p. 37).
Read more about this topic: Nil Ideal
Famous quotes containing the words relation to, relation and/or ideals:
“... a worker was seldom so much annoyed by what he got as by what he got in relation to his fellow workers.”
—Mary Barnett Gilson (1877–?)
“When needs and means become abstract in quality, abstraction is also a character of the reciprocal relation of individuals to one another. This abstract character, universality, is the character of being recognized and is the moment which makes concrete, i.e. social, the isolated and abstract needs and their ways and means of satisfaction.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“The measure discriminates definitely against products which make up what has been universally considered a program of safe farming. The bill upholds as ideals of American farming the men who grow cotton, corn, rice, swine, tobacco, or wheat and nothing else. These are to be given special favors at the expense of the farmer who has toiled for years to build up a constructive farming enterprise to include a variety of crops and livestock.”
—Calvin Coolidge (1872–1933)