Let $R$ be a Dedekind ring and $M$ is a finitely generate $R$-module. Let $T$ be the relation (the first syzygy module) of $M$. (T is the kernel of the canonical map $\eta : R^n \to M$ which sends the unit row $e_j$ on the generator $v_j$ of M for $1\leq j\leq n$). Using Structure theorem, we can write the $R$-module as $M= R/d_1\oplus \cdots \oplus R/d_r $, where $d_1, \ldots, d_r$ are ideals of $R$ subject to $0 \subseteq d_1 \subseteq d_2 \subseteq \cdots \subseteq d_r \subset R$. The ideals $d_1, \ldots, d_r$ are said to be the elementary divisor ideals of $M$.
How can I prove that the elementary ideals $\mathfrak{E}_i=\mathfrak{E}_i(M/R)= \prod_{j=1}^i d_j$ of $M$ are generated by the determinants of $(n-i)\times (n-i)$ minors of the $(n-i) \times n$ matrices formed from any $n-i$ rows of $\ker(\eta)$ (relation matrix $T$)?.
Convert $T$ in to the smith normal form (SNF), then we get a diagonal matrix say $\mathit{diag}(n_1,\ldots, n_2)$ with $n_{i}|n_{i+1}$. Then it holds that $\prod_{j=1}^in_i$ is the gcd of all $i\times i$ minors of $T$.
In that case the elementary divisor ideals are the principal ideals $d_i=n_{s+1-i}R$ $(1\leq i\leq s)$ generated by the elementary divisors $n_i$, and vice versa.