---- > [!definition] Definition. ([[submodule generated by a subset]]) > Let $R$ be a [[ring]] and $M$ an $R$-[[module]], and let $A \subset M$ be a subset of $M$. By the [[free module|universal property of free modules]], there is a unique [[linear map|homomorphism]] of $R$-[[module|modules]] $\varphi_{A}:R^{\oplus A} \to M$ > from the [[direct sum of modules|direct sum]] $R^{\oplus A}$ to $M$. > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAmAXVJADcBDAGwFcYkQBBEAX1PU1z5CKMsWp0mrdgB1pAC2YBzGAAJZAY2ZoefEBmx4CRMgAZxDFm0QgAsjv4GhRE6TM0LU6wCUAesFkQaCxwKhzcPOIwUMoIKKAAZgBOEAC2SC4gOBBIAMzuklYgAFb2IEmpuTRZSOT5ljLSDIloclil5WmIGdWItRL11rL4OPQgNIz0AEYwjAAKAobCIIlYinI4YyDTYFBIALQ5JtyU3EA > \begin{tikzcd} > R^{\oplus A} \arrow[r, "\varphi"] & M \\ > & \huge \cup \\ > & A \arrow[luu, "j"] \arrow[uu, "\iota"', bend right] > \end{tikzcd} > \end{document} > ``` > > $\im \varphi_{A}$ is a [[submodule]] of $M$, which we call the **submodule generated by $A$ in $M$** and denote by $\langle A \rangle$. Thus[^1], $\langle A \rangle=\left\{ \sum_{a \in A} r_{a} a : r_{a} \neq 0 \text{ for only finitely many elements } a \in A\right\}. $It is clear that $\langle A \rangle$ is the smallest [[submodule]] of $M$ containing $A$. > > If $M=\langle A \rangle$ for a finite set $A$, then we call $M$ **finitely generated**. Note that the condition $M=\langle A \rangle$ for finite $A$ is equivalent to stating there is a [[surjection|surjective]] $R$-[[linear map]] $R^{\oplus n} \twoheadrightarrow M$ > for some $n$, i.e., there exists an [[exact sequence]] $R^{\oplus n} \to M \to 0$ > for some $n$. > ^definition [^1]: Recalling how $\varphi$ is (how it must be!) defined. > [!note] Remark. > This justifies the notation $\langle \ \cdot \ \rangle$ for [[submodule generated by a subset]] in linear algebra. ^note > [!basicnonexample] Warning. > Finitely generated $R$-modules are really important, but not always well-behaved — for example, a [[submodule]] of a finitely generated module may not be finitely generated! Take $R=\mathbb{Z}[x_{1},x_{2},\dots]$, a [[polynomial 4|polynomial ring]] on infinitely many [[indeterminate|indeterminates]]. Then $R$ is finitely generated as an $R$-[[module]]: indeed, $1$ generates it. However, the [[ideal]] $(x_{1}, x_{2}, \dots)$ > of $R$ generated by all indeterminates is *not* finitely generated as an $R$-[[module]] (see Aluffi). ^warning ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` # Legacy span Let $V$ be a [[vector space]] over a [[field]] $\mathbb{F}$. # Definition The set of all [[linear combination]]s of a list of [[vector]]s $v_1,\dots,v_n$ in $V$ is called the [[span]] A list $a_{1},\dots,a_{n} \in$ $V$ is said to **span** $V$ if any [[vector]] $v \in V$ can be represented as $\begin{equation}c_1a_1 + \dots + c_ma_m, \end{equation}$ where $c_{1}, \dots, c_{m}$ are scalars in the ground [[field]] $F$. That is, $a_{1},\dots,a_{n}$ [[span]] $V$ if any vector $v\in V$ can be written as a [[linear combination]] of them. The set of all [[linear combination]]s of lists of [[vector]]s $a_1,\dots,a_m$ is called the **span** of $v_1, \dots, v_m$, denoted $\span(v_1, \dots, v_n.)$ In other words, $\span(v_1, \dots, v_m) = \lb a_1v_1 + \dots + a_mv_m : a_1, \dots, a_m \in \mathbb{F} \rb.$ # Results - [[span is the smallest containing subspace]]