Example of a simple module which does not occur in the regular module?

Question

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

Answer 1

If you consider the ring $A$ of matrices of the form $$\begin{pmatrix} a & b\\ 0& c\end{pmatrix}$$ then there are 2 simple $A$-modules, both 1 dimensional, one where the matrix above acts by $a$ and one where it acts by $c$.

Now, in any $A$-representation, a simple submodule is a vector sent to a multiple of itself by every element of $A$. For the left regular representation, the vectors that have this property are $$\begin{pmatrix} d & e\\ 0& 0\end{pmatrix}$$ and this is the sum of two copies of the same simple, where the left action just multiplies by $a$ and never $c$ (if you look at the right regular representation, you'll get the $c$-representation and not the $a$ instead). The $c$-representation is a quotient of the regular representation, but not a sub.

Answer 2

Take any commutative nonfield domain. A quotient by a maximal ideal is a simple module, but nonfield domains obviously do not have minimal right ideals.

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Rings $R$ for which simple right modules embed in $R_R$ are called right Kasch rings. Right Artinian local rings and commutative Artinian rings are examples of right Kasch rings, but as you see above domains are not Kasch on either side. More generally, any ring with right socle zero can't be right Kasch.

You can read a bit about then in Lam's Lectures on modules and rings.