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This definition can be applied in particular to square matrices. The matrix
is nilpotent because A3 = 0.
The nilpotent elements from a commutative ring form an ideal; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element in a commutative ring is contained in every prime ideal of that ring, and in fact the intersection of all these prime ideals is equal to the nilradical.
If x is nilpotent, then 1 − x is a unit, because xn = 0 entails
- (1 − x) (1 + x + x2 + ... + xn−1) = 1 − xn = 1.
Nilpotency in physics
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