Nilpotent Elements of Commutative Ring form Ideal
Jump to navigation
Jump to search
Theorem
Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$.
The subset of nilpotent elements of $R$ form an ideal of $R$.
Proof
Let $N$ be the subset of nilpotent elements.
Because $0_R$ is nilpotent, $0_R \in N$ and so $N$ is non-empty.
Let $x \in N$ and $a \in R$.
We have:
\(\ds \exists n \in \Z_{>0}: \, \) | \(\ds x^n\) | \(=\) | \(\ds 0_R\) | Definition of Nilpotent Ring Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^n \circ x^n\) | \(=\) | \(\ds 0_R\) | Definition of Ring Zero | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {a \circ x}^n\) | \(=\) | \(\ds 0_R\) | Power of Product of Commutative Elements in Semigroup | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a x\) | \(\in\) | \(\ds N\) | Definition of $N$ |
Let $x, y \in N$.
Let $x^n = 0$ and $y^m = 0$.
By the Binomial Theorem, $\paren {x - y}^p = 0$ for $p \ge n + m - 1$.
Thus $x - y \in N$.
Thus from Test for Ideal, $N$ is an ideal.
$\blacksquare$
Also see
Sources
- 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra: Chapter $1$: Rings and Ideals: $\S$ Nilradical and Jacobson Radical
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $11 \ \text {(ii)}$