 Finitely generated ideal

An ideal $I$ in a commutative ring $R$ is finitely generated if there are $a_1,\dots ,a_n\in I$ such thatThe 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 $2=2\cdot 6-10$ 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 $k[x_1,x_2,\dots ]$, 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.