One sided ideals of a semisimple ring.

Question

Let $A$ be a semisimple ring. I'm wondering whether all ideals of $A$ are two sided. I know that all semisimple rings are both left and right semisimple. And, since $A$ is a semisimple module over itself, we can write $$A=\bigoplus_{i=1}^n S_i $$ where $S_i=A/m_i$ for some maximal ideal. Thus, $$A=\prod_{i=1}^n A/m_i. $$ I believe that every two sided ideal should be a direct sum of the simple submodules, but what can be said of specifically one sided ideals? Since $A$ need not be commutative, I don't believe that every ideal necessarily is two sided, but I'm not sure how to go about finding the one sided ideals.

Answer 1

I'm not sure how to go about finding the one sided ideals.

By Artin-Wedderburn, $A\cong\prod_{i=1}^k M_{n_i}(D_i)$ for some positive integers $k, n_i$ and division rings $D_i$.

The right ideals of $A$ are products of right ideals of each factor.

The right ideals of a matrix ring can be completely described.

Putting these two things together, you can say you have a description of right ideals of $A$.

I don't believe that every ideal necessarily is two sided,

That would be correct. For example, $\begin{bmatrix}k&k \\ 0&0\end{bmatrix}$ is a proper right ideal of $M_2(k)$ which is not a left ideal.

In fact, the only semisimple rings in which one-sided ideals are all two-sided are finite products of division rings.