Are zero rings and modules semisimple?

Question

I tried searching in the web, but found little. An $R$-module is semisimple if it is a direct sum $\bigoplus_{i \in I} M_i$ of simple $R$-modules $M_i$. What are the conventions for the zero module? $\bigoplus_{i \in \varnothing} M_i = \{0\}$ and it's vacuously true that $M_i$ is simple for all $i \in \varnothing$. In particular, is a zero ring semisimple?

Answer 1

Yes. There is no need for a special convention: as you observed, the zero module satisfies the definition of a semisimple module and so it is semisimple. In particular, the zero ring is semisimple as a module over itself, and so is a semisimple ring.