Decomposition of $K[X]$-modules, where $K$ a field

Question

I saw something like "$K[X]/(X^2) \cong K[X]/(X) \oplus K[X]/(X)$ is not true because otherwise $K[X]/(X^2)$ would be annihilated by the action of X". What does it mean exactly? Also, I saw $R^2 \cong R[X]/(X-1)\oplus R[X]/(X-1)$ is not true because that would mean $X-1$ acts by zero on $R^2$...I don't understand what it means...

Answer 1

If $R$ is a ring and $V$ is an $R$-module, it is standard to say that:

• An element $v\in V$ is annihilated by $r\in R$ if $rv=0$

• The action of $r\in R$ is the multiplication operator $v\mapsto rv$ of $V$.

Variations abound; We say that "$V$ is anihhilated by the action of $r\in R$" to mean that $rv=0$ for each $v\in V$;

The annihilator of $V$ is the ideal $Ann(V)$ of $R$ given by $Ann(V)=\left\{r\in R:rv=0\text{ for all }v\in V\right\}$. We could also say that "$Ann(V)$ is the largest ideal of $R$ which annihilates all of $V$".

Anyway, that is what the nomenclature means. Of course these "annihilation" properties are preserved under isomorphism: If $\phi\colon V\to W$ is an $R$-module isomorphism and $v\in V$ is annihilated by $r\in R$, then $\phi(v)$ is annihilated by $r$ as well.

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Now note that $K[X]/(X)$, seen as a $K[X]$-module in the standard manner, is annihilated by the element $X$ of $K[X]$; Thus the same is true for the direct sum: $K[X]/(X)\oplus K[X]/(X)$ is annihilated by the element $X$ of $K[X]$.

Any $K[X]$-module isomorphic to $K[X](X)\oplus K[X]/(X)$ is thus also annihilated by $X$. But $K[X]/(X^2)$ is not annihilated by $X$, so it cannot be isomorphic to $K[X]/(X)\oplus K[X]/(X)$.