Irreducible module
I've met the following definition of irreducible module: an $R$ module $M$ is said to be irreducible if it contains no proper submodules: in other words, if $N \subset M$ is a submodule than either $n=0$ or $N=M$. Form the other side, one can think of module map $R \times M \to M$ as a representation $\pi\colon R \to \operatorname{End}(M)$ (where $M$ is viewed as an abelian group) via the formula $\pi(r)m:=rm$. Suppose that $R$ and $M$ are also vector spaces (over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$). My question is the following: does the notion of irreducibility coincide with the following property: a representation $\pi$ is irreducible? The latter would mean that the only module maps $T:M \to M$ are of the form $cI_{M}$ where $c \in \mathbb{K}$
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$\begingroup$The two things are not equivalent. There are modules which are irreducible in the first sense and not in the second.
For example, let R be a division ring and M=R. As soon as R contains your field strictly, there are non-scalar endomorphisms.
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