One example of a nilpotent element is a nilpotent matrix.
2.
A nilpotent element in a nonzero ring is necessarily a zero divisor.
3.
This is primarily because the commutativity assumption ensures that the product of two nilpotent elements is again nilpotent.
4.
A nilpotent element is an element a such that a ^ n = 0 for some n > 0.
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Compares several different definitions of finite W-algebras, which are certain associative algebras associated to nilpotent elements of semisimple Lie algebras.
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The converse is clear : an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal.
7.
The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal.
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For example, there exist nilradical of a ring, the set of all nilpotent elements, need not be an ideal unless the ring is commutative.
9.
Equivalently, the radical of " I " is the pre-image of the ideal of nilpotent elements ( called nilradical ) in R / I.
10.
The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero.
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