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.

Methodology/Instructions

• 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.

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