Idempotent and Nilpotent Element of a Ring

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

Definition of an Idempotent Element

Let R be a ring. An element e in R is called an idempotent element if e² = e. For example, the zero element 0 is an idempotent element as 0² = 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.

1. The zero element 0 and the multiplicative identity I are idempotent elements of R.
2. Let e be an idempotent element in R. Then e² = e. Note that (I-e)² = (I-e) (I-e) = I² – Ie – eI + e² = 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.

1. $\bar{2}$ is a nilpotent element in the ring Z₄. Because, $\bar{2}^2 =\bar{0}$.
2. $\bar{4}$ is a nilpotent element in the ring Z₁₆. This is because $\bar{4}^3=\bar{64} =\bar{0}$.

Related Topics: Introduction to Ring Theory

Units of a Ring

Characteristic of a Ring

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 aⁿ = 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

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