Coherent ring

Remarks

Equivalently, for every $a_1, \dots, a_n \in R$, the module of relations $\set{(r_1, \dots, r_n) \in R^n : r_1 a_1 + \cdots + r_n a_n = 0}$ is finitely generated. Example: every Noetherian ring is coherent, because the module of relations is a submodule of $R^n$, which is a Noetherian module. Non-example: let $R = k[x, y_1, y_2, \dots]/(x y_1, x y_2, \dots)$ over a field $k$. The relations of the single element $x$ form the ideal $(y_1, y_2, \dots)$ of $R$, which is not finitely generated. So the ideal $(x)$ is finitely generated but not finitely presented. For a noncommutative ring one defines left coherent rings: every finitely generated left ideal is finitely presented.