homomorphic image of semi-simple ring is not necessarily semi-simple

Question

I am studying non-commutative algebra and i have the following exercise:

Proving a homomorphic image of semi-simple ring is also a semi-simple ring.

I have try to search it in the internet but I meet some remark in some book that this problem is not true. In Certain Number-Theoretic Episodes In Algebra, you can see the Observation 14.2.4 page 492, they say that a homomorphic image of semi-simple ring is also a semi-simple ring. But I think the exercise they gave not true because $\mathbb{Z}$ is not semi-simple

I wonder if this problem is True? Please give me some hint. If it is false, give me some counter example.

Thanks!

Answer 1

The problem is probably that they use semisimple to mean

trivial Jacobson radical

This is a convention in some algebra books. Some authors go out of the way to emphasize “semisimple Artinian” to refer to what is called “semisimple” nowadays because of this problem. And I have also seen trivial Jacobson radical rings relabeled as "J-semisimple," or alternatively semiprimitive (which I think is what Jacobson himself called such rings.)

The example given would indeed fit the usage of semisimple to mean $J(R)=\{0\}$. For the other meaning of "semisimple," it is true that every quotient is also semisimple, so there would be no counterexample.