Lattice-ordered matrix algebras over real GCD-domains

Abstract

Let $R\subset \mathbb{R}$ be a GCD-domain. In this paper, Weinberg's conjecture on the $n\times n$ matrix algebra $M_n(R)(n\ge 2)$ is proved. Moreover, all the lattice orders (up to isomorphisms) on a full $2\times 2$ matrix algebra over $R$ are obtained.

9 pages, this is the final version (to appear in the Journal: Communications in Algebra). Thanks to the anonymous referee's helpful suggestions, we replace the UFD rings by the GCD-domains in this paper