Summary of "Abstract Algebra | The characteristic of a ring."

Main Ideas and Concepts

Characteristic of a Ring:
The Characteristic of a Ring \( R \) is defined as the smallest positive integer \( n \) such that adding the element \( r \) to itself \( n \) times yields zero. If no such \( n \) exists, the characteristic is said to be zero.
Zero Divisors:
Elements \( a \) and \( b \) in a ring \( R \) are called Zero Divisors if neither is zero but their product is zero.
Integral Domain:
A commutative ring with a multiplicative identity and no Zero Divisors is called an Integral Domain.
Examples of Characteristics:
- The characteristic of the Integers \( \mathbb{Z} \), Rationals \( \mathbb{Q} \), Real Numbers \( \mathbb{R} \), Complex Numbers \( \mathbb{C} \), and polynomial rings \( \mathbb{Z}[X] \) is zero.
- The characteristic of the ring \( \mathbb{Z}/n\mathbb{Z} \) (denoted \( \mathbb{Z}_n \)) is \( n \).
Standard Results:
- If \( R \) is a ring with characteristic \( n \), then adding the element \( 1 \) to itself \( n \) times results in zero.
- If \( R \) is an Integral Domain, its characteristic must either be zero or a prime number.

To determine the Characteristic of a Ring:
- Check if adding an element \( r \) to itself \( n \) times results in zero.
- Identify the smallest \( n \) for which this is true.
- If no such \( n \) exists, conclude that the characteristic is zero.
Example Calculation:
For \( \mathbb{Z}_6 \) (the Integers modulo 6):
- The elements are \( \{0, 1, 2, 3, 4, 5\} \).
- Adding \( 2 \) to itself three times gives \( 6 \equiv 0 \mod 6 \), hence the characteristic is \( 3 \).
Proving Characteristics:
For a ring \( R \) with characteristic \( n \):
- Show that \( n \) is the least number such that \( n \times 1 = 0 \).
- If \( n \) is composite, demonstrate that it leads to the existence of Zero Divisors, contradicting the Integral Domain property.