Block Decomposition

$R$ be a ring with unity.

Lemma

Let $E$ be a simple $R$-module. If $E^n=E^m$, then $n=m$.

Proof

Let $D=En{d}_R(E)$. By the Schur’s lemma, $D$ is a division ring. So,

$$M_n(D){\cong }_{Ring}En{d}_R(E^n){\cong }_{\text{Ring or mod}}En{d}_R(E^m){\cong }_{Ring}{M}_m(D).$$

Therefore, $n^2=m^2$ implies $n=m$.

Proposition

If $\phi :E_1^{n_1}\oplus \cdots \oplus E_r^{n_r}\to F_1^{m_1}\oplus \cdots \oplus F_s^{m_s}$ is an isomorphism with $E_i$ and $F_j$ are simple and each $E_i$‘s and $F_j$‘s are non-isomorphic. Then $r=s$ and $\exists \sigma \in S_r$ such that $E_i{\cong }_{Mod}{F}_j{\cong }_{Mod}{F}_{\sigma (i)}$ and $n_i=m_j=m_{\sigma (i)}$.

Proof

Let $\phi =\left({\phi }_{ji}:E_i\to F_j\right)$ be the isomorphism. Since it is injective $\forall i\exists j$ such that ${\phi }_{ji}\ne 0$. Then by the Schur’s lemma, $E_i{\cong }_{Mod}{F}_j$. Hence, for each $i$, uniquely exists $j$ such that ${\phi }_{ji}\ne 0$. Otherwise, $F_j\cong F_{j^{\prime }}$. Also, since ${\phi }^{-1}$ is an isomorphism. Therefore, there exists a permutation $\sigma \in S_r$.

Corollary

If $E$ is a finite direct sum of simple modules, then the isomorphism class of simple components of $E_i$, and multiplicities are well-defined.

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From the proposition, ${\phi }_{ji}:E_i^{n_i}\to F_{\sigma (i)}^{m_{\sigma (i)}}$ is a block diagonal matrix of $\phi$. For example,

$$\left[\begin{array}{cccc}{M}_1& 0& \cdots & 0\\ 0& M_2& \cdots & 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& M_r\end{array}\right]$$

where $M_i$ is a map between ${\phi }_{ji}:E_i^{n_1}to{F}_j^{m_j}$.

Therefore, we can call $\phi$ is an block decomposition.

For the Semi Simple Ring

Now, we can decompose semi simple ring $R$, using the proof of AW, as $R{=}_{R-mod}\oplus B_i$ where $B_i={\sum }_{\text{Simple left ideal}L{\cong }_{R-mod}{L}_i}L$.

In a mathematical language, this is the theorem¹.

Let $R$ be semisimple ring, and let $E$ be an $R$-module $\ne 0$. Then $E={\oplus }_{i=1}^s{R}_{i}E={\oplus }_{i=1}^s{e}_{i}E,$

and $R_{i}E$ is the submodule of $E$ consisting of the sum of all simple submodules isomorphic to $L_i$ where $L_i$, $R_i$, and $e_i$ are the things in Artin-Weddenburn Theorem context.

Corollary

1. Let $R$ be semi simple ring. Then every simple module is isomorphicc to one of the simple left ideals $L_i$.
2. A simple ring has exactly one simple module, up to isomorphism. So that $M{\cong }_{R-mod}\oplus L$.

Non Trivial Central Idempotent

Different Point of View, it is same as finding orthogonal non trivial central idempotents.

External and Internal Decomposition

External

If there are rings $R_1,\dots ,R_s$, we can construct explicitly as a product ring $R_1\times \cdots \times R_s$.

As a block decomposition

$$R_1\times \cdots \times R_s=R_1\times 0\times \cdots \times 0\oplus 0\times R_2\times \cdots \times 0\oplus 0\times \cdots R_s.$$

Internal

Let $R$ be a ring, then we have a block decomposition $R=B_1\oplus \cdots \oplus B_S$ where $B_i$‘s are 2-sided ideals. Also, as we proved in Artin-Weddenburn Theorem, $B_i=(R{e}_i,e_i)$ is a ring itself. However it is not a subring of R because it has different unity.

So as a ring, it is not a direct sum of rings, it is a coproduct in ring category.

Footnotes

1. Theorem 4.4 in Lang page. 653 ↩