Algebra
《Algebra》 · 第 220 页 · Chapter IV，Section 6「Modules over a Principal Ideal Domain」，Theorem 6.4 及其后定名段
Theorem 6.4. Let $A$ be a left module over an integral domain $R$ and for each $a\in A$ let $\mathcal O_a=\{r\in R\mid ra=0\}$. (i) $\mathcal O_a$ is an ideal of $R$ for each $a\in A$. (ii) $A_t=\{a\in A\mid\mathcal O_a\ne0\}$ is a submodule of $A$. (iii) For each $a\in A$ there is an isomorphism of left modules $R/\mathcal O_a\cong Ra=\{ra\mid r\in R\}$. Let $R$ be a principal ideal domain and $p\in R$ a prime. (iv) If $p^ia=0$ (equivalently $(p^i)\subset\mathcal O_a$), then $\mathcal O_a=(p^j)$ with $0\le j\le i$. (v) If $\mathcal O_a=(p^i)$, then $p^ja\ne0$ for all $j$ such that $0\le j<i$. […] Let $A$ be a module over an integral domain. […] The submodule $A_t$ in Theorem 6.4 is called the torsion submodule of $A$. $A$ is said to be a torsion module if $A=A_t$ and to be torsion-free if $A_t=0$.