Basic Definitions

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  • Definition of a commutative ring

A ring [math]R[/math] is a commutative ring if [math]ab=ba[/math] for all [math]a, b \in R[/math].

  • Definition of an integral domain

An integral domain is a commutative ring [math]R[/math] with [math]0 \neq 1[/math] such that whenever [math]ab=0[/math] then either [math]a = 0[/math] or [math]b=0[/math].

  • Definition of a field

An field is a commutative ring [math]F[/math] with [math]0 \neq 1[/math] such that every nonzero element [math]a \in F[/math] has an inverse in [math]F[/math], that is, there is an element [math]a^{-1} \in F[/math] such that [math]aa^{-1} = 1 = a^{-1}a.[/math]

  • Definition of a subring

A subset [math]S[/math] of a ring [math]R[/math] is a subring of [math]R[/math] if

(1) [math]1 \in S;[/math]

(2) [math]a - b \in S[/math] for all [math]a, b \in S;[/math]

(3) [math]ab \in S[/math] for all [math]a, b \in S.[/math]

  • Definition of a subfield

A subring [math]S[/math] of a field [math]F[/math] is a subfield of [math]F[/math] if every nonzero element of [math]S[/math] has an inverse in [math]S.[/math]

  • Definition of characteristic

Let [math]R[/math] be a ring. If there is a positive integer [math]n[/math] such that [math]n1 = 0[/math], then the least such integer is called the characteristic of [math]R[/math]. If no such integer exists, then we say the characteristic of [math]R[/math] is zero.

  • Theorem

Let [math]R[/math] be an integral domain. Then the characteristic of [math]R[/math] is either zero or a prime.

Next section on Homomorphisms and Ideals

Back to Chapter 1: Basics of Ring Theory