Nilpotent element

In mathematics, an element, x, of a ring, R, is called nilpotent if there exists some positive integer, n, such that xⁿ = 0.

The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.

No nilpotent element can be a unit (except in the trivial ring {0}, which has only a single element 0 = 1). All non-zero nilpotent elements are zero divisors.

An n-by-n matrix A with entries from a field is nilpotent if and only if its characteristic polynomial is tⁿ.

If x is nilpotent, then 1 − x is a unit, because xⁿ = 0 entails

More generally, the sum of a unit element and a nilpotent element is a unit when they commute.

The nilpotent elements from a commutative ring ${\displaystyle R}$ form an ideal ${\displaystyle {\mathfrak {N}}}$; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element ${\displaystyle x}$ in a commutative ring is contained in every prime ideal ${\displaystyle {\mathfrak {p}}}$ of that ring, since ${\displaystyle x^{n}=0\in {\mathfrak {p}}}$. So ${\displaystyle {\mathfrak {N}}}$ is contained in the intersection of all prime ideals.

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