Commutative ring/Ideal/Finitely generated/Definition

Generated ideal

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

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

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