In this article, we will learn about idempotent element and nilpotent element of a ring with examples.

Table of Contents

## Definition of an Idempotent Element

Let R be a ring. An element e in R is called an idempotent element if e^{2} = e. For example, the zero element 0 is an idempotent element as 0^{2} = 0.

## Examples of Idempotent Elements

Let R be a ring with unity I. The the examples of the idempotent elements in R are given as follows.

- The zero element 0 and the multiplicative identity I are idempotent elements of R.
- Let e be an idempotent element in R. Then e
^{2}= e. Note that (I-e)^{2}= (I-e) (I-e) = I^{2}– Ie – eI + e^{2}= I – 2e + e = I-e. This show that I-e is an idempotent element of R.

## Definition of an Nilpotent Element

Let R be a ring. An element a in R is called a nilpotent element if a^{k} = 0, the zero element, for some integer k. For example, 0 is a nilpotent element.

## Examples of Nilpotent Elements

A list of few examples of nilpotent elements in a ring are given below.

- $\bar{2}$ is a nilpotent element in the ring Z
_{4}. Because, $\bar{2}^2 =\bar{0}$. - $\bar{4}$ is a nilpotent element in the ring Z
_{16}. This is because $\bar{4}^3=\bar{64} =\bar{0}$.

**Related Topics:** Introduction to Ring Theory

## Solved Problems

**Question 1:** Prove that a ring R with no zero divisors does not have any nilpotent element.

**Answer:**

If possible, suppose that a be a non-zero nilpotent element in R. Then a^{n} = 0 for some integer n>1. This gives us that

a ⋅ a^{n-1} = a^{n-1} ⋅ a = 0

⇒ a is a zero divisor of R.

This proves that a is a nilpotent element implies a is a zero divisor. In other words, a ring R having no zero divisors contains no nilpotent element.

**Also Read:** Zero divisors of a ring

## FAQs

**Q1: What are idempotent elements of a ring?**

Answer: If an element a in a ring R satisfies a^{2}=a, then it is called an idempotent element. For example, the zero 0 in R is an idempotent element.

**Q2: What are nilpotent elements of a ring?**

Answer: If an element a ∈ R satisfies a^{n}=0, then it is called a nilpotent element. For example, 0 is a nilpotent element.