Past exam of the commutative algebra course of the University of Oxford / 2023 / 2 / d / Solution

Use the direct product of ringswith coordinatewise addition and multiplication and unit $(1,1,\dots )$. Define the shiftCoordinatewise operations show that $g$ preserves addition, multiplication and the unit, so it is a ring endomorphism. It is surjective because $(0,b_0,b_1,\dots )$ maps to any prescribed $(b_0,b_1,\dots )$. Its kernel of a ring homomorphism isso it is not injective. This is the noninjective surjective shift on a product of fields.

This ring fails the Noetherian hypothesis in c. Indeed, let $e_n$ have a single $1$ in coordinate $n$. The ideals $I_n=(e_0,\dots ,e_n)$ form the strictly ascending chainsince $e_{n+1}\notin I_n$.