We look at a pricipial ideal domain $R$. And a $R$ Module $M$, we define $T$ as the torisionmodule of $M$ with $T=\{a\in M ; \exists \alpha \in R-\{0\}:\alpha a=0\}$. Becuase $R$ is a pricipila ideal domaine this is also a submodule of $M$.
I have a question about a specific part in the proof of a theorem:
We want to write a $R$-Module $M$ which has a finite generating system as direct sum of $T$ and another module $F$ which is finite and free, ie $F$ is isomorphic to $R^d$. Furthermore there are non-units $\alpha_1,...,\alpha_n\in R-\{0\}$ such that $\alpha_j|a_{j+1}$ and
$T$ is isomorph to a (constructed) direct sum of the form $\bigoplus_{j=1}^{n}R/\alpha_jR$.
Also $\alpha_j$ is unique (except for associativity) and $d$ is unique.
I am only interested in the existence proof for now
We know since $M$ is finitely constructed that there exists a map $\phi:R^r\rightarrow M$ with $e_i\mapsto z_i$ and where $z_i$ is a generating system, $\phi$ is an epimorphism.
Because of the fundamental theorem of homorphism there exists a bijective map between $M$ and $R^r/ker \phi$ Because $\ker \phi \subset R^r$ we can use the theorem for elemntary divisors and find a basis $x_1,...,x_r$ of $R^r$ and elements $\alpha_1,...,\alpha_n\in R-\{0\}$ such that $\alpha_1x_1,...,\alpha_nx_n$ is a base of $\ker \phi$ and $\alpha_|\alpha_{i+1}$. We define $\alpha_{n+1},...,\alpha_r=0$.
The map
$\psi: R^r=\bigoplus_{j=1}^{r}Rx_j\rightarrow \bigoplus_{j=1}^{r}Rx_j/R\alpha_jx_j$ with $(\gamma_1,...,\gamma_r)\mapsto (\bar{\gamma_1},...,\bar{\gamma_r)}$.
$\bar{\gamma_j}$ is the residue class of $\gamma_j$ in $Rx_j/R\alpha_jx_j$
is an epimorphism
We have $\ker \phi=\ker\psi$
$\equiv$ means that there is a isomorphism. Due to the fundamental theorem of homomorphisms we have
$$M\equiv R^r/ker \phi = (\bigoplus_{j=1}^{n}Rx_j)/\ker \psi\equiv \bigoplus_{j=1}^{r}Rx_j/R\alpha_jx_j\equiv R^{r-n}\oplus\bigoplus_{j=}^{n}R/\alpha_jR$$
What I don't understand is that the author says $T$ corresponds to (there is an isomorphism with) $\bigoplus_{j=}^{n}R/\alpha_jR$
Can someone explain?