On Galois theory of division rings

On the Galois Theory of Division Rings

1. Throughout this paper, K will represent a division ring and L a galois division subring. We are interested in establishing a galois theory for the extension K/L when K/L is locally finite. In order to do this one must identify the galois subrings of K containing L. An example given by Jacobson [4] shows that not every such division subring is galois. However, we obtain that each subring subj...

Triangulaizability of Algebras Over Division Rings

On nest modules of matrices over division rings

Let $ m , n in mathbb{N}$, $D$ be a division ring, and $M_{m times n}(D)$ denote the bimodule of all $m times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We ...

Galois Theory and Integral Models of Λ-rings

We show that any Λ-ring, in the sense of Riemann–Roch theory, which is finite étale over the rational numbers and has an integral model as a Λ-ring is contained in a product of cyclotomic fields. In fact, we show that the category of them is described in a Galois-theoretic way in terms of the monoid of pro-finite integers under multiplication and the cyclotomic character. We also study the maxi...

Generalized affine transformation monoids on Galois rings

Let A be a ring with identity. The generalized affine transformation monoid Gaff(A) is defined as the set of all transformations on A of the form x → xu + a (for all x ∈ A), where u,a∈ A. We study the algebraic structure of the monoid Gaff(A) on a finite Galois ring A. The following results are obtained: an explicit description of Green’s relations on Gaff(A); and an explicit description of the...