= Solution
Use the <direct product of rings>
$$
T=\prod_{n\geq0}\mathbb R,
$$
with coordinatewise addition and multiplication and unit $(1,1,\ldots)$. Define the shift
$$
g(a_0,a_1,a_2,\ldots)=(a_1,a_2,a_3,\ldots).
$$
Coordinatewise operations show that $g$ preserves addition, multiplication and the unit, so it is a <ring endomorphism>. It is <surjective> because $(0,b_0,b_1,\ldots)$ maps to any prescribed $(b_0,b_1,\ldots)$. Its <kernel of a ring homomorphism> is
$$
\boxed{\ker g=\{(a,0,0,\ldots):a\in\mathbb R\}\ne0,}
$$
so it is not <injective>. This is the <noninjective surjective shift on a product of fields>.
This <ring> fails the <Noetherian> hypothesis in <2/c>. Indeed, let $e_n$ have a single $1$ in coordinate $n$. The <ideals> $I_n=(e_0,\ldots,e_n)$ form the strictly ascending chain
$$
I_0\subsetneq I_1\subsetneq I_2\subsetneq\cdots,
$$
since $e_{n+1}\notin I_n$.
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