Open up a book on abstract algebra, and you're likely to see the following definition:

Definition of submodule generated by a subset

Suppose $M$ is an $R$-module and $X$ is any collection of elements in $M$. The submodule of $M$ generated by $X$ is the collection of all finite $R$-linear combinations of elements in $X$. In other words, it is the subset of all elements of the form $\underset{x\beta \x88\x88X}{\beta \x88\x91}{r}_{x}x$, where ${r}_{x}\beta \x88\x88R$, all but finitely many of which are zero.

This submodule is usually denoted $RX$, which is simultaneously understandable and horrible. It is the smallest submodule of $M$ that contains the elements of $X$.

Of course, we should officially verify that the set $RX$ is indeed a submodule of $M$, and that it truly is the smallest submodule of $M$ that contains $X$. This would be relatively straightforward, but there is an alternate definition of $RX$ that makes all of those properties immediate.

Consider the following equivalent definition, which uses our free module construction. By the universal property of the free module $F(X)$, the set inclusion $i:X\beta \x86\x92U(M)$ corresponds to a module morphism $f:F(X)\beta \x86\x92M$. The image of this morphism is a submodule of $M$ that contains $X$. By the construction of the free module $F(X)$, this image is exactly the set $RX$ described above. Furthermore, if $N$ is any submodule of $M$ that contains $X$, then the inclusion $j:X\beta \x86\x92U(N)$ corresponds to a module morphism $g:F(X)\beta \x86\x92N$, and again by the definition of $F(X)$ the image of this morphism is $RX$. Thus $RX$ is a submodule of $N$, making it the smallest submodule of $M$ that contains $X$.

Generators for a given submodule

If $N$ is a submodule of $M$, we say a subset of elements $X$generates N if $N=RX$. In this case, we call $X$ a set of of generators (or a generating set) for $N$, and we say $N$ is generated by $X$. In terms of elements, $N$ is generated by $X$ exactly when every $n\beta \x88\x88N$ can be expressed in the form $n={\displaystyle \underset{x\beta \x88\x88X}{\beta \x88\x91}{r}_{x}x}$ for some ${r}_{x}\beta \x88\x88R$ (all but finitely many zero); note that such an expression does not have to be unique.

If there exists a finite set $X$ that generates $N$, then we say $N$ is finitely generated.

If there exists a singleton set that generates $N$, then we say $N$ is cyclic. In this case, this is equivalent to the existence of a single element ${n}_{0}\beta \x88\x88N$ such that every element $n\beta \x88\x88N$ can be expressed in the form $n=r{n}_{0}$ for some $r\beta \x88\x88N$; as before, this expression does not have to be unique.

Connecting everything back to free modules once more, we see that $N$ is generated by $X$ exactly when the module morphism $F(X)\beta \x86\x92N$ (corresponding to the inclusion $X\beta \x86\x92U(N)$) is surjective. The submodule $N$ is cyclic exactly when there is a surjection $F(\{\beta \x88\x99\})\beta \x86\x92N$, where $\{\beta \x88\x99\}$ is a/the singleton set.

A word on relations

A common way to describe a group is to give a presentation, which consists of a list of generators together with their "fundamental" relations. We can make that concept clear in the context of modules.

Suppose $M$ is a module. A subset ${X}_{1}$ of $M$ is a set of generators exactly when the corresponding module morphism ${\mathrm{{\rm O}\x80}}_{1}:F({X}_{1})\beta \x86\x92M$ is surjective. Moreover, the kernel of this morphism captures all of the relations between those generators, as it consists exactly of every possible $R$-linear combination of the generators that equals zero (in $M$). We can now say that a set of "fundamental" relations on the generators is precisely a set ${X}_{2}$ of generators for $\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})$. As before, this corresponds to a surjective module morphism $F({X}_{2})\beta \x86\x92\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})$.

Composing this new module morphism with the inclusion of $i:\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})\beta \x86\x92F({X}_{1})$, we then have a chain of morphisms

with $\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})=\mathrm{im}(i\beta \x88\x98{\mathrm{{\rm O}\x80}}_{2})$. This is the start of an exact sequence of morphisms, something to be explored later.