Finite Dimensional Simple Modules

Question

What does mean by finite dimensional simple module? Generally simple module does not havy any non zero proper submodule.So for that reason I think always simple modules must be 1- dimensional. But this is wrong, I cant find any mistake. Also If M is simple then $ M = m R$ for some $ m \in M $. It implies $ M $ is 1 dimensional. Where am I wrong?

Answer 1

One says "finite dimensional module" for a module over an algebra over a field $k$. The dimensionality is that of the $k$ vector space structure of the module.

The dimension of a simple module is quite often not $1$.

For example, if you have a $\mathbb Q$ vector space $V$ of countably infinite dimension, and $R=\mathrm{End}(V_k)$, then $R$ is a $\mathbb Q$ algebra with infinite dimensional simple module $V_R$.

Even when $R$ and $V$ are finite dimensional, the dimension of a simple module does not have to be $1$. For example, $\mathbb C$ is a $2$-dimensional $\mathbb R$ algebra, and a simple module over itself.

As you say, though, a simple module is always finitely generated, in fact always a cyclic module. The issue is that the minimum number of $R$ generators does not have to match the minimum number of $k$ generators.