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Preface. Let $D$ be a division ring, let $R=\mathbb{M}_n(D)$. Let $\sideset{_R}{}V$ be the (unique) simple left $R$-module. This module can be given as $V=D^n$ the collection of $n$-tuple columns, with action by $R$ on the left given by matrix multiplication.

Observation 1. $End(\sideset{_R}{}V)\cong D$.

Observation 2. The commutative group $V$ can also be considered as a left $D$-module, with action of $D$ given by componentwise multiplication. Then $End(\sideset{_D}{}V)\cong R$.

Observation 3. Intuitively, it seems that if $R$ is an arbitrary ring, and $\sideset{_R}{}V$ is its arbitrary left module, then $End(\sideset{_R}{}V)$ is a subring of $End(V)$ - the ring of endomorphisms of the group $V$. Furthermore, intuitively, "the larger" the ring $R$, the smaller $End(\sideset{_R}{}V)$, since an endomorphism $e\in End(\sideset{_R}{}V)$ must "respect more scalars from $R$" - observations 1 and 2 support that, for example. So, it almost seems like some relation of the sort $End(\sideset{_R}{}V)\approx End(V)/R$ should be in place.

Is there a simple way to express the endomorphism ring of a left $R$-module $End(\sideset{_R}{}V)$ in terms of the ring $End(V)$, or is this a meaningless idea and I am missing something?

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The $R$-module structure on $V$ corresponds to a ring morphism $f: R\to End(V)$. Then $End(_R V)$ is the centralizer of the image of $f$ in $End(V)$.

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