Commutative ring/Ideal/Finitely generated/Definition

For a family of element ${\displaystyle {}a_{1},a_{2},\ldots ,a_{n}\in R}$ in a commutative ring ${\displaystyle {}R}$, we denote by ${\displaystyle {}(a_{1},a_{2},\ldots ,a_{n})}$ the ideal generated by these elements. It consists of all linear combinations

${\displaystyle r_{1}a_{1}+r_{2}a_{2}+\cdots +r_{n}a_{n},}$

where ${\displaystyle {}r_{1},r_{2},\ldots ,r_{n}\in R}$.