Characteristic of a Ring: Definition, Examples

A positive integer n is said to be the characteristic of a ring R, if it is the least number such that na = 0 for all a ∈ R. In this article, we will study the characteristic of a ring along with its definition, examples, and theorems.

Definition

The characteristic of a ring R is the least positive integer n such that na = 0 for all a ∈ R. It is usually denoted by char R.

If such an integer n does not exist, then char R = 0.

Examples

1. The characteristic of the ring (ℤ, +, ⋅) is 0.
2. char ℝ = 0
3. The characteristic of the ring (ℤ_n, +, ⋅) is n. Therefore, char ℤ₄ = 4.

Theorems

Theorem: Let R be a ring with unity I. Then char R = n if n is the least positive integer such that nI = 0.

Proof:

For a ∈ R, we have

na = a + a + … + a (n times)

⇒ na = a⋅(I + I + … + I)

⇒ na = a⋅nI

⇒ na = a⋅0 as we are given that nI=0

⇒ na = 0.

Thus, we have shown that na=0 for all a in R. As nI=0 holds for the least positive integer, we conclude that na=0 ∀ a ∈ R holds where n is the least. This proves that char R = n.

Now, if no such n with nI =0 exists, then there will not be any n for which na=0 ∀ a ∈ R holds. Hence, char R = 0.

Read Also: An Introduction to Ring Theory

Units of a Ring: Definition, Examples, How to Find

Properties

1. char R = 1 if and only if R is a zero/null ring.
2. If n is the least positive integer with nI=0, then char R = n, where I denotes the unity.

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