Let \(\varphi\colon \mathbb{Z}\to \mathbb{Z}_{4}\times \mathbb{Z}_{6}\), \(\varphi(n)=(n\bmod 4,\ n\bmod 6)\). How many elements does the image of \(\varphi\) have?
The kernel is \(4\mathbb{Z}\cap 6\mathbb{Z}=12\mathbb{Z}\), so the image is isomorphic to \(\mathbb{Z}/12\mathbb{Z}\), with 12 elements. It is not the whole product, which has 24.
If \(\varphi\colon R\to S\) is a ring homomorphism with kernel \(\ker\varphi\), the quotient ring of \(R\) by \(\ker\varphi\) is isomorphic to the image of \(\varphi\).
\[R/\ker\varphi\;\cong\;\varphi(R)\]
More practice problems: first isomorphism theorem for rings