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**stephi85** Hi everyone. I'm preparing for a test tomorrow and there are concepts and methods, which I seem to have forgotten or never understood properly.

For eksample there are these exercises, which I'm having difficulties with, if someone could help me throught them, it would be nice (Nod) :

1) Explain why the $\displaystyle \mathbb{Z}$-modules $\displaystyle \mathbb{Z}/8\mathbb{Z}$ and $\displaystyle \mathbb{Z}/36\mathbb{Z}$ have finite lenght and write a chain of composition.

Now, in 1) I would say that $\displaystyle 8=2^3$ and therefore

$\displaystyle (0)\subset 4\mathbb{Z}/8\mathbb{Z}\subset 2\mathbb{Z}/8\mathbb{Z} \subset \mathbb{Z}/8\mathbb{Z}$ correct?

and $\displaystyle 36=2^2 3^2$ so $\displaystyle \mathbb{Z}/36\mathbb{Z}\cong \mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ so $\displaystyle (0)\subset \mathbb{Z}/2\mathbb{Z} \subset \mathbb{Z}/3\mathbb{Z} \subset \mathbb{Z}/36\mathbb{Z}$? So thus the lenght is three in both cases and thus finite?

well, they're both finite modules and thus obviously Artinian (and Noetherian). so they have finite length. your solution for $\displaystyle \mathbb{Z}/36 \mathbb{Z}$ is wrong. Quote:

4) Are some of the three $\displaystyle k[X, Y]$-modules

$\displaystyle k[X, Y]/(XY)$ , $\displaystyle k[X, Y]/(X^2)$ , $\displaystyle k[X, Y]/(Y^2)$ isomorphic? Are some of them isomorphic as rings?

In (4) I would say that the two last are isomorphic, because you can make an ismomorphism from $\displaystyle k[X, Y]/(X^2)$ to $\displaystyle k[X, Y]/(Y^2)$ by sending X to Y, Y to X and 1 to 1. Correct?

the only possible isomorphism here is that "as rings" we have $\displaystyle k[X,Y]/(X^2) \cong k[X,Y]/(Y^2).$ the isomorphism sends $\displaystyle X$ to $\displaystyle Y$ and $\displaystyle Y$ to $\displaystyle X.$ to see that there are no other isomorphisms let