Homomorphisms and Ideals

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  • Definition of a ring homomorphism

Let [math]R, S[/math] be rings. A map [math]\varphi : R \to S[/math] is a ring homomorphism if, for all [math]a, b \in R,[/math]

(1) [math]\varphi(a+b) = \varphi(a)+\varphi(b);[/math]

(2) [math]\varphi(ab) = \varphi(a)\varphi(b);[/math]

(3) [math]\varphi(1) = 1.[/math]

  • The image of a ring homomorphism

The image Im$\varphi$ of a ring homomorphism [math]\varphi : R \to S[/math] is defined as

Im$\, \varphi = \{s \in S \ |\ s = \varphi(r) \mbox{ for some } r \in R\}.$

The ring homomorphism [math]\varphi[/math] is onto if and only if Im$\,\varphi = S.$

If [math]\varphi : R \to S[/math] is a ring homomorphism then Im$\varphi$ is a subring of $S$.

  • Definition of an ideal

A subset [math]I[/math] of a ring [math]R[/math] is an ideal of [math]R[/math] if

(1) [math]0 \in I;[/math]

(2) [math]a_1 - a_2 \in I[/math] for all [math]a_1, a_2 \in I;[/math]

(3) [math]ar \in I[/math] and [math]ra \in I[/math] for all [math]a \in I[/math] and [math]r \in R.[/math]

Note that if [math]R[/math] is a commutative ring, then we may replace (3) by (3'):

(3') [math]ra \in I[/math] for all [math]a \in I[/math] and [math]r \in R.[/math]

  • The kernel of a ring homomorphism

The kernel Ker$\varphi$ of a ring homomorphism [math]\varphi : R \to S[/math] is defined as

Ker$\, \varphi = \{r \in R \ |\ \varphi(r) = 0\}.$

The ring homomorphism [math]\varphi[/math] is one-to-one if and only if Ker$\,\varphi = \{0\}.$

If [math]\varphi : R \to S[/math] is a ring homomorphism then Ker$\,\varphi$ is an ideal of $R$.

  • Definition of a principal ideal

Let [math]R[/math] be a commutative ring and let [math]a \in R[/math]. The set [math]\{ra : r \in R\}[/math] is an ideal of [math]R[/math] and is called the principal ideal of [math]R[/math] generated by [math]a[/math]. This ideal is denoted by [math](a)[/math].

An ideal [math]I[/math] of [math]R[/math] is a principal ideal of [math]R[/math] if there is some [math]a \in I[/math] with [math]I = (a).[/math]

  • Theorem

(1) Every ideal of [math]{\mathbb Z}[/math] is a principal ideal.

(2) Let [math]F[/math] be a field. Then every ideal of [math]F[x][/math] is a principal ideal.




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Back to Chapter 1: Basics of Ring Theory