 Noninjective surjective shift on a product of fields

For a field $k$, the shift on the direct product of rings ${\prod }_{n\ge 0}k$ preserves coordinatewise addition, multiplication and the unit. It is a surjective ring endomorphism because an arbitrary first coordinate can be prepended to a desired output. Its kernel of a ring homomorphism consists of the sequences supported in coordinate zero, so it is not injective. The ring is not Noetherian: the ideals generated by the first $n+1$ coordinate idempotents form a strictly ascending chain.