Nilpotent

An element

x

of a ring is nilpotent when

x^{n} = 0

n

Square-zero element ( $a^{2} = 0$ )

An element of an associative algebra is square-zero if its square vanishes. Every such element is a nilpotent element. These elements are preserved by an algebra isomorphism, so their structure distinguishes algebras. Their set need not be a vector space: the matrix units

E_{12}

and

E_{21}

are square-zero, while their sum has square equal to the identity in

M_{2} (K)

. In a square-zero ideal, by contrast, every pairwise product vanishes and all its elements are square-zero. In

K [t] / (t^{3})

the square-zero elements are exactly the multiples of

t^{2}

, over every field:

(y t + z t^{2})^{2} = y^{2} t^{2}

y = 0

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In mathematics, particularly in the study of linear algebra and abstract algebra, the term "nilpotent" refers to a specific type of element in a ring or algebra. An element $ a $ of a ring $ R $ is said to be nilpotent if there exists a positive integer $ n $ such that \[ a^n = 0. \] In this context, $ 0 $ represents the additive identity in the ring $ R $.