Confusion about the definition of semisimple rings

Question

A module is semisimple if it is a direct sum of simple submodules, and a (unital) ring is semisimple if it is semisimple as a right module over itself. Right submodules of a ring are in particular right ideals. This, however, confuses me. Suppose that $I_1,...,I_n\subset R$ are minimal right ideals, so they are simple, such that $R=I_1\oplus ... \oplus I_n$. Does this have the usual direct sum structure? Because if so, the unit of $R$ is $(1,...,1)$ which means each ideal contains $1$, so they are in particular the ring itself!

Answer 1

You are confusing the internal direct sum of right ideals with the direct product of rings.

There is no reason to say “$(1,1,1,..)$ is the unit”. All you know is $1=\sum i_j$ where $i_j\in I_j$. It is true that $i_j$ is a generator of $i_jR$, but it does not have an identity necessarily, and it doesn’t contain $1$.

I the direct product of rings, it is true that $1=\sum e_j$ where $e_j$ is the identity of the ring $R_j$, but $e_j$ is not the same thing as $1$, and while $R_j$ is an ideal in the product, it does not contain $1$.