= Finitely generated ideal
{title2=$I=(a_1,\ldots,a_n)$}
An <ideal> $I$ in a <commutative ring> $R$ is finitely generated if there are $a_1,\ldots,a_n\in I$ such that
$$
I=(a_1,\ldots,a_n)=\left\{\sum_{j=1}^nr_ja_j:r_j\in R\right\}.
$$
The generators need not generate $I$ as an additive group; arbitrary ring coefficients are allowed. The zero ideal is generated by zero, or by the empty list under that convention.
For example, $(6,10)=(2)$ in $\mathbb Z$: both generators are divisible by two, while $2=2\cdot6-10$ gives the reverse inclusion. In $k[x,y]$, the <ideal> $(x,y)$ consists of <polynomials> with zero constant term.
Finite generation is not automatic in every <ring>. In $k[x_1,x_2,\ldots]$, the ideal of all polynomials with zero constant term is not finitely generated. Any finite list involves only finitely many variables; setting those variables to zero kills the generated ideal but leaves an unused variable nonzero. A <Noetherian ring> is exactly a commutative ring in which every ideal is finitely generated.
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