Basis of a module

A basis $S$ of an $R$-module $M$ is a subset such that every element of $M$ has a unique expression as a finite $R$-linear combination of elements of $S$.

For an element $m={\sum }_{e\in E}{r}_{e}e$ of a free module with specified basis of a module $E$, its coordinate support is the finite set of basis of a module elements having nonzero coefficients. This is well-defined by uniqueness of the expression. The coordinate support of a sum is contained in the union of the two coordinate supports, and scalar multiplication cannot enlarge it. It is distinct from the prime-ideal-valued support of a module.