Let $ R \subset \R $ be a GCD-domain. In this paper, Weinberg's conjecture on the $ n \times n $ matrix algebra $ M_{n}(R) \ (n \geq 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