# Endomorphism ring of a module in terms of endomorphism ring of the underlying group

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?

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 $$End(V)$$.