Ring Homomorphisms and Ideals

For any integer \(n\) we can define a ring homomorphism \(\phi : {\mathbb Z} \rightarrow {\mathbb Z}_n\) by \(a \mapsto a \pmod{n}\). This is indeed a ring homomorphism, since \[\begin{aligned}\phi( a + b ) & = (a + b) \pmod{n} \\ & = a \pmod{n} + b \pmod{n} \\ & = \phi( a ) + \phi(b)\end{aligned}\] and \[\begin{aligned}\phi( a b ) & = ab \pmod{n} \\ & = a \pmod{n}\cdot b \pmod{n} \\ & = \phi( a ) \phi(b)\end{aligned}\]. The kernel of the homomorphism \(\phi\) is \(n {\mathbb Z}\).

Let \(C[a, b]\) be the ring of continuous real-valued functions on an interval \([a,b]\) as in . For a fixed \(\alpha \in [a, b]\), we can define a ring homomorphism \(\phi_{\alpha} : C[a, b] \rightarrow {\mathbb R}\) by \(\phi_{\alpha} (f ) = f( \alpha)\). This is a ring homomorphism since \[\begin{aligned}\phi_{\alpha}( f + g ) = (f + g)( \alpha) = f(\alpha) + g(\alpha) = \phi_{\alpha}( f ) + \phi_{\alpha}(g ) \\ \phi_{\alpha}( f g ) = (f g)( \alpha) = f(\alpha) g(\alpha) = \phi_{\alpha}( f ) \phi_{\alpha}(g )\end{aligned}\]. Ring homomorphisms of the type \(\phi_{\alpha}\) are called evaluation homomorphisms.

Every ring \(R\) has at least two ideals, \(\{ 0 \}\) and \(R\). These ideals are called the trivial ideals.