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.
Table of Contents
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
- The characteristic of the ring (ℤ, +, ⋅) is 0.
- char ℝ = 0
- The characteristic of the ring (ℤn, +, ⋅) 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
- char R = 1 if and only if R is a zero/null ring.
- If n is the least positive integer with nI=0, then char R = n, where I denotes the unity.
FAQs
Answer: If na = 0 for all a ∈ R, then n is called the characteristic of R if it is the least positive integer. For example, the characteristic of the ring (ℤ5, +, ⋅) is 5.
