Does a matrix ring over a ring satisfy the Koethe conjecture if the coefficient ring itself satisfies the Koethe conjecture?

Question

I just want to know whether the following statement is true or false.

If $R$ is a ring satisfying the Koethe conjecture, then the matrix ring over $R$ also satisfies the Koethe conjecture.

Or is it equivalent to the Koethe conjecture itself?

I will be grateful if you can link to a paper or a book.