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Let $R = k[x, y]$, the polynomial ring in two variables, where $k$ is an algebraically closed field. Let $a, b, c, d\in k[x^2, y^2] \subset R$, and $$I = (a, b, c + dx) \triangleleft R$$ and $$J = I y + (a + bx) \triangleleft R.$$ Then $J \subset I$ and $yI \subset J$. Thus, the $R$-module $$I/J$$ has a $k[x]$-module structure induced by its $R$-module structure. I wonder how to decompose $I/J$ as a $k[x]$-module in terms of the invariant factors, according to the structure theorem of finitely generated module over p.i.d?

Some general ideas of how to approach this problem would also be helpful. Thanks!

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    Can you show that $I/J$ is finitely generated $K[X]$-module? – 2017-02-19
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    Yes, $I$ is a finitely generated $k[x, y]$-module, so is its quotient $I/J$. But the $k[x, y]$-module structure on $I/J$ induces a $k[x]$-module structure, since $y I \subset J$. Therefore, it is a finitely generated $k[x]$-module. – 2017-02-19
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    Also, notice that as a $k[x]$-module, $I/J$ is generated by $b$ and $c + dx$. So basically the $k[x]$-module structure is determined by the kernel of the canonical projection $k[x] \oplus k[x] \rightarrow k[x].b + k[x].(c +dx)$, which should be a free $k[x]$-module of rank 2. I have hard time calculating this kernel. – 2017-02-19

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