free module

《代数学引论》 · 第 167 页 · 定义 7

若 $R$-模 $M$ 有一基，则 $M$ 叫做一个自由 $R$-模。

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《Abstract Algebra》 · 第 354 页 · Section 10.3「Generation of Modules, Direct Sums, and Free Modules」，Definition

An $R$-module $F$ is said to be free on the subset $A$ of $F$ if for every nonzero element $x$ of $F$, there exist unique nonzero elements $r_1,r_2,\ldots,r_n$ of $R$ and unique $a_1,a_2,\ldots,a_n$ in $A$ such that $x=r_1a_1+r_2a_2+\cdots+r_na_n$, for some $n\in\mathbb Z^+$. In this situation we say $A$ is a basis or set of free generators for $F$.