Artin-Wedderburn theorem contradiction? Mind = Blown

[Silversonic]

Homework Statement

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.

Homework Equations

Relevant theorems

http://img713.imageshack.us/img713/8471/g7pn.png

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 that's 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?

 

[mighty2000]

I would first like to commend you for your thorough and thoughtful approach to solving this problem. It is clear that you have a strong understanding of the relevant theorems and concepts. However, I believe the issue here is a misunderstanding of what is meant by a direct product of rings.

In your first approach, you correctly identify that R is the direct sum of the simple ideals S1, S2, and S3. However, this does not mean that R is the direct product of these three rings. The direct sum and direct product are two different operations, and the direct product is typically used when dealing with modules rather than rings.

In your second approach, you correctly identify that R can be written as a direct product of three rings, each isomorphic to a division ring. However, these rings are not the same as the simple ideals S1, S2, and S3. In fact, the rings M1(D1) and M2(D2) are matrix rings over division rings, whereas S1, S2, and S3 are simple ideals of R.

So, in summary, R can be written as a direct product of matrix rings over division rings, but these are not the same as the simple ideals S1, S2, and S3. The image you provided is showing that R can be written as a direct product of two matrix rings over division rings, not three.

I hope this helps clarify the issue for you. Keep up the good work in your studies of rings and modules!