Factorization in hereditary orders

Abstract

Let A denote a central separable algebra over a global field K with ring of integers R, and let Λ denote an R-order in A. It is shown that when Λ is hereditary and A is Eichler/ R, then λϵΛ has a left factor having reduced norm an associate of rϵR whenever r divides the reduced norm of λ. In particular, if A= M n ( K), it follows that a matrix in a hereditary R-order in a A having determinant divisible by r∈ R has a left factor having determinant r.