0

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.

Ty Perkins
  • 610
  • 1
    No, it's not true that all ideals are two-sided. For example, $A = M_n(k)$ has interesting left and right ideals (hint: it has a natural action on $k^n$, pick any nonzero vector and consider the kernel of the action on this vector). – Qiaochu Yuan Sep 19 '23 at 22:51
  • @QiaochuYuan In your example, $A$ is certainly simple, but is it necessarily semisimple? – Ty Perkins Sep 19 '23 at 23:01
  • 2
    Yes. Every ring which is both simple and artinian is semisimple. By Artin-Wedderburn these are exactly the rings of the form $M_n(D)$ where $D$ is a division ring. This ring has a unique simple module, namely $D^n$, and every module is a direct sum of copies of it. This is actually enough to tell you what the left and right ideals are, using the fact that a left resp. right ideal is a submodule with respect to left resp. right multiplication. – Qiaochu Yuan Sep 19 '23 at 23:03

1 Answers1

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.

rschwieb
  • 153,510