An example of a simple module which does not occur in the regular module? Let $K$ be a field and $A$ be a $K$-algebra.
I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem  $A$ , as a left regular module, can be written as a direct sum of indecomposable $A$-modules, that is 
$A=\oplus_{i=1}^n S_i$ where each $S_i$ is indecomposable $A$-module
Moreover, each $S_i$ contains only one maximal submodule, which is given by $J_i= J(A)S_i$, and every simple $A$-module is isomorphic to some $A/J_i$.
My question is that, can you,  please,  tell me an example of a non simisimple algebra, or a ring, such that it has a simple module which does not occur in the regular module.
By occur I mean it has to be isomorphic to a simple submodule of a regular module

The simple modules for $F[x]$ are $F[x]/(p)$ where $p=p(x)$ is irreducible. These do not occur as submodules of $F[x]$.
