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.
- 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?
- 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?