On Galois matrix extensions of high order

Abstract

Let R be a commutative ring with 1 and A a central R-Galois algebra with an inner Galois group G of order n for some integer n. Let M_m(R) be the ring of m × m-matrices over R for an integer m. Then M_m(R) is also a central Galois R-algebra with an inner Galois group for each m = n⁽²ⁱ⁾ where i is a non-negative integer. In particular, 2 is invertible in R if and only if M_m(R) is a central Galois R-algebra with an inner Galois group for each m = 2⁽²ⁱ⁾ where i is a non-negative integer.