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:
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“The difference between objective and subjective extension is one of relation to a context solely.”
—William James (1842–1910)
“It does not follow, because our difficulties are stupendous, because there are some souls timorous enough to doubt the validity and effectiveness of our ideals and our system, that we must turn to a state controlled or state directed social or economic system in order to cure our troubles.”
—Herbert Hoover (1874–1964)