# Homework Help: Artin-Wedderburn theorem contradiction? Mind = Blown

1. Nov 23, 2013

### Silversonic

1. The problem statement, all variables and given/known data

My question basically wants me to write the direct product of rings $R = \mathbb{Z}_3 \oplus \mathbb{Z}_5 \oplus \mathbb{Z}_5$ as a direct product of matrix rings over division rings.

2. Relevant equations

Relevant theorems

http://img713.imageshack.us/img713/8471/g7pn.png [Broken]

3. The attempt at a solution

So I found two ways of doing this which seemed contradictory to the above Theorem.

First, $R$ is the direct sum of the simple ideals $(\mathbb{Z}_3, 0 , 0) = S_1$, $(0, \mathbb{Z}_5, 0) = S_2$ and $(0, 0, \mathbb{Z}_5) = S_3$. These are simple since the $\mathbb{Z}_3, \mathbb{Z}_5$ are fields (and have no proper non-zero ideals).

But then treat $R$ as a left $R$ module. Then these three simple ideals above become three simple submodules of $R$ and

$R = S_1 + S_2 + S_3$

As left $R$ modules, $S_2 \cong S_3$ whereas $S_1$ is not isomorphic to either of the other two (different cardinalities).

Then $R \cong S_1 \oplus S_2 \oplus S_3 \cong S_1 \oplus {S_2}^2$

$End_R(M)$ is the set of R-module homomorphisms from $M$ to $M$. Another theorem previously shows as rings $R$ is isomorphic to $End_R(R)$. But since this is the case, Theorem 4.24 tells me;

$R \cong End_R(R) \cong End_R(S_1 \oplus {S_2}^2) \cong M_1(D_1) \oplus M_2(D_2)$

Where $D_1 = End_R(S_1), D_2 = End_R(S_2)$ are division rings.

So thats one direct product of matrix rings over division rings. But also I could've simply said.

$\mathbb{Z_3} \cong M_1(\mathbb{Z}_3), \mathbb{Z_5} \cong M_1(\mathbb{Z}_5)$ as rings.

$R \cong M_1(\mathbb{Z}_3) \oplus M_1(\mathbb{Z}_5) \oplus M_1(\mathbb{Z}_5)$

Also a direct product of matrix rings over fields (thus division rings)

But this surely contradicts what is said at the very bottom of my image above. Since one is a direct product over 2 matrix rings, another is a direct product over 3? So can anyone tell me where I'm wrong?

Last edited by a moderator: May 6, 2017