Let R be a commutative ring and let A be any ideal of R....

Let $R$ be a commutative ring and let $A$ be any ideal of $R$. Show that the nil radical of $A, N(A)=\left\{r \in R \mid r^{n} \in A\right.$ for some positive integer $n(n$ depends on $r)\}$, is an ideal of $R$. $[N(\langle 0\rangle)$ is called the nil radical of $R .]$

Key Concepts

Binomial Expansion in Commutative Rings

The binomial expansion, which holds in commutative rings due to the commutativity of multiplication, allows one to expand powers of sums. This tool is frequently employed to demonstrate that the sum of elements with nilpotent properties also satisfies a nilpotency condition, thereby establishing closure under addition—a key property for an ideal.

Nilpotent Element

A nilpotent element in a ring is one that, when raised to some positive integer power, results in the zero element (or, more generally, falls into a specific ideal). The concept of nilpotency is crucial in defining radicals and understanding certain structural properties of rings.

Nil Radical

The nil radical of an ideal A is the collection of all elements in the ring such that some power of each element is in A. This concept generalizes the idea of nilpotency and is important in ring theory because it identifies a subset of elements having this nilpotent behavior relative to an ideal, and it often reveals deep structural properties of the ring.

Commutative Ring

A commutative ring is an algebraic structure in which addition and multiplication are defined and behave similarly to those operations on integers, with the key property that multiplication is commutative. This property is essential in many proofs involving ideals and nilpotency because it allows for techniques like the binomial expansion to be applied.

Ideal

An ideal is a special subset of a ring that is closed under addition and under multiplication by any element of the ring. Ideals serve as the building blocks for constructing quotient rings and for understanding the structure of rings, making them central to many arguments, including those about radical ideals.

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