Simple Ring

If we consider ring $R$ as a left $R$-module, then $R$-submodule $N$ is a left ideal of $R$. Similarly, If $N$ is left and right $R$-submodule, then $N$ is a two sided ideal of $R$.

Definition

Let $R$ be a semi simple ring. If every simple summands of $R$ as a $R$-module are isomorphic, then $R$ is a simple ring.

Example

For example, let $R=M_2(\mathbb{H})$, where $\mathbb{H}$ is a quaternion algebra over $\mathbb{R}$. Then $M_2\left(\mathbb{H}\right)$ is a simple ring such that

$$\left(\begin{array}{cc}\ast & 0\\ \ast & 0\end{array}\right)\oplus \left(\begin{array}{cc}0& \ast \\ 0& \ast \end{array}\right).$$

Remark

Note that $M_2\left(\mathbb{H}\right)\times \mathbb{R}$ is semi simple ring because there is a $0\times \mathbb{R}$ factor in the summand which is not isomorphic to the others.

Characterization

Artin-Weddenburn Theorem

$R$ is a simple ring if and only if $R$ is artinian ring and has no proper 2-sided ideals.

Ideal

A left ideal $I$ of $R$ is called simple if it is simple as a module.