= Cyclic module
{title2=$M=Rm$}
{wiki}
A <module> is cyclic if one element $m$ generates it: every element is $rm$ for some scalar $r\in R$. The generating element need not give unique coefficients. The <module homomorphism> $R\to M$, $r\mapsto rm$, is <surjective>. Over a <commutative ring>, its <kernel> is the <annihilator> $\operatorname{Ann}_R(m)$, giving
$$
M\cong R/\operatorname{Ann}_R(m).
$$
Thus <cyclic modules> are exactly the <quotient modules> $R/I$ for <ideals> $I$ in $R$. For a left <module> over a <noncommutative ring>, use a <left ideal> and interpret the quotient as a <module>, with no claim that it is a <quotient ring>. See https://faculty.niu.edu/math_beachy/aaol/modules.shtml[John A. Beachy's Abstract Algebra Online, section 10.1].
The <module> $\mathbb Z/6\mathbb Z$ is generated by $\overline1$, and also by $\overline5$, whose successive multiples are $\overline0,\overline5,\overline4,\overline3,\overline2,\overline1$. The element $\overline2$ generates only $\{\overline0,\overline2,\overline4\}$, a <submodule> with three elements. A cyclic <module> can be infinite and free, as $\mathbb Z=\mathbb Z\cdot1$ shows, or finite and torsion, as $\mathbb Z/6\mathbb Z$ shows. One generator counts directions of generation, not the number of elements and not freedom from relations.
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