= Noninjective surjective shift on a product of fields
{title2=$g((a_n)_{n\geq0})=(a_{n+1})_{n\geq0}$}
For a <field> $k$, the shift on the <direct product of rings> $\prod_{n\geq0}k$ 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.
\Comment[[
Exact target: nilradical-commutes-with-localization
Existing parent: Nilradical
Intended file: /home/ciro/bak/git/codex-wiki/commutative-algebra.bigb
]]
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