In general the infinite direct sum of a simple module $M$ is an essential in the infinite direct product of $M$. But if $M$ is a module over semisimple ring then the infinite direct sum most be direct summand of the infinite direct product, which means that the infinite direct sum is equal to infinite direct product!!!!!
How can that be?
Thanks for any help.
No, it is not. Over a semisimple ring, every module is semisimple, and semisimple modules have no nontrivial essential submodules.
This "fact" in the highlighted box above is apparently the problem.
Yes, it is an infinite direct sum of simple $R$ modules, but there is no reason it is equal to the sum of copies of $M$ constructed earlier. It can have more members in the sum.
Perhaps if you considered the field of two elements $F_2$ for a moment, and the vector spaces $\oplus_{i\in \omega} F_2$ and $\prod_{i\in \omega}F_2$. The first one is a countable dimensional vector space, and the second one is an uncountable dimensional vector space, isomorphic to $\oplus_{i\in I}F_2$ for some uncountable index set $I$.