Nilpotent - Wikipedia
WEBIn mathematics, an element of a ring is called nilpotent if there exists some positive integer, called the index (or sometimes the degree), such that =. The term, along with its sister idempotent , was introduced by Benjamin Peirce in the context of his work on the classification of algebras.
Nilpotent ideal - Wikipedia
WEBIn mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I k = 0. [1] By I k, it is meant the additive subgroup generated by the set of all products of k elements in I. [1]
Nilradical of a ring - Wikipedia
WEBIn algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: It is thus the radical of the zero ideal. If the nilradical is the zero ideal, the ring is called a reduced ring. The nilradical of a commutative ring is …
Nilpotent Elements in Rings - Mathonline - Wikidot
WEBAn element $a \in R$ is said to be Nilpotent if there exists an $n \in \mathbb{N}$ such that $a^n = 0$. By definition, the additive identity $0$ is always a nilpotent element in a ring $(R, +, *)$ .
Nilpotent ideal - Encyclopedia of Mathematics
WEBNov 24, 2018 · In the ring of upper-triangular matrices over a field the matrices with 0's along the main diagonal form a nilpotent ideal. Every element of a nilpotent ideal is nilpotent. Every nilpotent ideal is also a nil ideal and is …
WEBIn a commutative ring, if u is a unit and b is nilpotent, the sum u + b is a unit: u + b = u(1 + b=u), and if b k = 0 then an elementary calculation shows 1 + b=u is a unit with inverse being the nite geometric series
Comm. Algebra - Nilpotent Elements - Stanford University
WEBNilpotent Elements. Let R R be a ring. We say x ∈ R x ∈ R is a zero divisor if for some nonzero y ∈ R y ∈ R we have xy = 0 x y = 0. Example: 2 is a zero divisor in Z4 Z 4. 5,7 are zero divisors in Z35 Z 35. A nonzero ring in which there are no nonzero zero divisors is called an integral domain.
Nilpotent element - Encyclopedia of Mathematics
WEBJul 11, 2022 · In a commutative ring $ A $ an element $ a $ is nilpotent if and only if it is contained in all prime ideals of the ring. All nilpotent elements form an ideal $ J $, the so-called nil radical of the ring; it coincides with the intersection of all prime ideals of $ A $.
WEBDefinition 4.6. Let R be a ring, and let N(R) = X {I : I is nilpotent ideal of R}, i.e. let N(R) be the sum of all nilpotent ideals of R. The ideal N(R) is called the nilradical of R. Clearly N(R) ⊆ X {I : I is nilpotent left ideal of R}. But the reverse inclusion also holds since every nilpotent left ideal I is contained in a nilpotent ...
WEBThroughout this paper, all rings are associative rings with 1. It is well known that a polynomial f(x) over a commutative ring is nilpotent if and only if each coe cient of f(x) is nilpotent. But this result is not true in general for noncommutative rings. For example, let R= M n(k), the n nfull matrix ring over some ring k6= 0.
Skip to content