linear extension

Let $R$ be a commutative ring, $M$ a free $R$-module, $B$ a basis of$M$, and $N$ a further $R$-module. Each element $m\in M$ then has aunique representation

$$m=\sum _{b\in B}{m}_{b}b,$$

where $m_b\in R$ for all $b\in B$, and only finitely many $m_b$ arenon-zero. Given a set map $f_1:B\to N$ we may therefore define the $R$-module homomorphism ${\phi }_1:M\to N$, called the linear extension of $f_1$, such that

$$m\mapsto \sum _{b\in B}{m}_b{f}_1(b).$$

The map ${\phi }_1$ is the unique homomorphism from $M$ to $N$ whose restriction to $B$ is $f_1$.

The above observation has a convenient reformulation in terms of category theory. Let $\mathrm{𝖱𝖬𝗈𝖽}$ denote the category of $R$-modules, and $\mathrm{𝖲𝖾𝗍}$ the category of sets. Consider the adjoint functors $U:\mathrm{𝖱𝖬𝗈𝖽}\to \mathrm{𝖲𝖾𝗍}$, the forgetful functor that maps an $R$-module to its underlying set, and $F:\mathrm{𝖲𝖾𝗍}\to \mathrm{𝖱𝖬𝗈𝖽}$,the free module functor that maps a set to the free $R$-module generated by that set. To say that $U$ is right-adjoint to $F$ is the same as saying that every set map from $B$ to $U(N)$, the set underlying $N$, corresponds naturally and bijectively to an $R$-module homomorphism from $M=F(B)$ to $N$.

Similarly, given a map $f_2:B^2\to N$, we may define thebilinear extension

${\phi }_2:$$M^2\to N$$(m,n)$$\mapsto {\displaystyle \sum _{b\in B}}{\displaystyle \sum _{c\in B}}{m}_b{n}_c{f}_2(b,c),$

which is the unique bilinear map from $M^2$ to $N$ whose restrictionto $B^2$ is $f_2$.

Generally, for any positive integer $n$ and a map $f_n:B^n\to N$, we may define the $n$-linear extension

${\phi }_n:$$M^n\to N$$m$$\mapsto {\displaystyle \sum _{b\in B^n}}{m}_b{f}_n(b)$

quite compactly using multi-index notation: $m_b=\prod _{k=1}^n{m}_{k,b_k}$.

Usage

The notion of linear extension is typically used as amanner-of-speaking. Thus, when a multilinear map is definedexplicitly in a mathematical text, the images of the basis elementsare given accompanied by the phrase “by multilinear extension” orsimilar.