Ring of functions
= Ring of functions
{title2=$k^X$}
For a <commutative ring> $k$ and a <set> $X$, the set $k^X$ of all <functions> $X\to k$ is a <commutative ring> under pointwise addition and multiplication. It is the <direct product of rings> with one copy of $k$ for each $x\in X$.
If $k$ is a <field> and $X$ is finite, every <ideal> is generated by some of the singleton <indicator functions> $e_x$. Indeed, whenever $f$ in the <ideal> has $f(x)\ne0$, multiplication by $e_x/f(x)$ puts $e_x$ in the <ideal>. Every <function> in that <ideal> is a finite sum of its coordinate multiples of these $e_x$.
If $X$ is infinite and $k$ is a nonzero <commutative ring>, the <functions> with finite <support> form an <ideal> without a finite <generating set of an ideal>. A finite collection of such <functions> vanishes outside one finite subset, as does the <ideal> it generates, whereas a singleton <indicator function> outside that subset belongs to the <ideal> of all finitely supported <functions>. Thus, for a <field> $k$, $k^X$ is a <Noetherian ring> exactly when $X$ is finite.