# Confusion about the definition of semisimple rings

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!

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 $$1$$.