The free module on a set and its standard basis

For a unital ring $R$ and a set $X$, the free left $R$-module on $X$ is $R^{(X)}:={⨁}_{x\in X}R\mathrm{.}$ For $x\in X$, the standard basis vector $e_x$ has coordinate $1_R$ at $x$ and zero elsewhere. Every element is represented by a unique finitely supported coefficient family $(r_x{)}_{x\in X}$, and is written ${\sum }_{x\in X}{r}_x{e}_x\mathrm{.}$ The standard basis map $X\to R^{(X)}$ sends $x$ to $e_x$. It is injective exactly when $1_R\mathrm{\ne }0$ or $X$ has at most one element; in these cases it is the standard basis inclusion. Indeed, for distinct $x,y$ the $x$-coordinates of $e_x$ and $e_y$ are $1_R$ and $0$. For the zero ring every $e_x$ is zero, but the indexed family $(e_x{)}_{x\in X}$ still has unique finitely supported coefficients.

More generally, a family $(b_x{)}_{x\in X}$ is a basis of a module $M$ when every element of $M$ is uniquely a finite $R$-linear combination of the $b_x$ (Generated submodule, cyclic and finitely generated modules, module basis and free module). For $X=\mathrm{\varnothing }$, $R^{(X)}=0$ and its empty family is a basis.