Homomorphisms and Ideals

• Definition of a ring homomorphism

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

(1) $\varphi(a+b) = \varphi(a)+\varphi(b);$

(2) $\varphi(ab) = \varphi(a)\varphi(b);$

(3) $\varphi(1) = 1.$

• The image of a ring homomorphism

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

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

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

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

• Definition of an ideal

A subset $I$ of a ring $R$ is an ideal of $R$ if

(1) $0 \in I;$

(2) $a_1 - a_2 \in I$ for all $a_1, a_2 \in I;$

(3) $ar \in I$ and $ra \in I$ for all $a \in I$ and $r \in R.$

Note that if $R$ is a commutative ring, then we may replace (3) by (3'):

(3') $ra \in I$ for all $a \in I$ and $r \in R.$

• The kernel of a ring homomorphism

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

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

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

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

• Definition of a principal ideal

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

An ideal $I$ of $R$ is a principal ideal of $R$ if there is some $a \in I$ with $I = (a).$

• Theorem

(1) Every ideal of ${\mathbb Z}$ is a principal ideal.

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

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