finitely generated module

A module $X$ over a ring $R$ is said to be finitely generated if there is a finite subset $Y$ of $X$ such that $Y$ spans $X$. Let us recall that the span of a (not necessarily finite) set $X$ of vectors is the class of all (finite) linear combinations of elements of $S$; moreover, let us recall that the span of the empty set is defined to be the singleton consisting of only one vector, the zero vector $\overrightarrow{0}$. A module $X$ is then called cyclic if it can be a singleton.

Examples. Let $R$ be a commutative ring with 1 and $x$ be an indeterminate.

1. $Rx=\{rx\mid r\in R\}$ is a cyclic $R$-module generated by $\{x\}$.

2. $R\oplus Rx$ is a finitely-generated $R$-module generated by $\{1,x\}$. Any element in $R\oplus Rx$can be expressed uniquely as $r+sx$.

3. $R[x]$ is not finitely generated as an $R$-module. For if there is a finite set $Y$ $R[x]$, taking $d$ to be the largest of all degrees of polynomials in $Y$, then $x^{d+1}$ would not be in the of $Y$, assumed to be $R[x]$, which is a contradiction. (Note, however, that $R[x]$ is finitely-generated as an $R$-algebra.)