# Difficulties with “old” definitions

I have a paper by Jacobson

Nathan Jacobson.Structure of Rings, Volume 37, Part 1. American Math-ematical Soc., revised edition, 1956.

Which really uses definitions that seem very complicated for what should be simple things. Any help to find simpler ways of defining the following or they're modern equivalents would be helpful.

1. Centralizer of a module.

We have an $A$-module $M$ and $E$ it's ring of endomorphisms. He definies the centraliser of the module $M$ as a ring $S$ which is anti-isomorphic to $E$. What is the point? basically it is just the ring of endomorphisms right?

1. reduced order of a group $G$ of automorphisms of a division ring $D$

he defines it as the product of the index of the subgroup $H\subset G$ of inner automorphisms by the dimensionality of the centralizer of a subring $E\subset D$ over the center. Is there another way of seing this?

• Does he use right modules or left modules? (That might explain the opposite ring puzzle.) – rschwieb Mar 7 '18 at 15:05
• Are you sure he says "the centralizer of a module is a ring isomorphic to $E^{op}$? That doesn't sound likely... What ring of endomorphisms? $A$-linear endomorphisms? or group endomorphisms? – rschwieb Mar 7 '18 at 15:10
• 1) right module. 2) I quote: "To achieve this we introduce a ring $\Gamma$ which is anti-isomorphic to the ring of endomorphisms of $M$ [...]. From now on, we shall call $\Gamma$ the centraliser of the module $M$. – tomak Mar 7 '18 at 15:29
• At times like this I really wish i had Structure of Rings :/ Hard to work out what could be meant without more context. There is perhaps some benefit of thinking about things in terms of centralizers in algebras rather than endmorphism rings. Multiplication on the right by elements of $A$ gives rise to group endomorphisms of $M$. The $A$-linear endomorphisms are precisely the centralizer in the full ring of group endomorphisms of that set of group endomorphisms. But I can't really see a use for using the opposite ring here... – rschwieb Mar 7 '18 at 15:41
• Another question linked to this, if we have a ring extension $D/E$ he says in his book that non-zero elements of the centraliser (i.e. the ring of endomorphisms) determine inner automorphisms which leave the elements of $E$ invariant. How do we get an inner automorphism? – tomak Mar 8 '18 at 10:29