Let \(\varphi\colon \mathbb{Z}_{12}\to \mathbb{Z}_{4}\) be reduction modulo 4, which is surjective. To which ring is \(\mathbb{Z}_{12}/\ker\varphi\) isomorphic?
By the first isomorphism theorem the quotient is isomorphic to the image, and the image is all of \(\mathbb{Z}_{4}\) because the map is surjective. The quotient has 4 elements, not 3 or 12.
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