On GL₂(R) Where R is a Boolean Ring

Extract

In this paper we characterize the 2 × 2 invertible matrices over a Boolean ring, and, using this characterization, show that every invertible matrix has order dividing 6. This suggests that GL₂ of a Boolean ring is built up out of copies of the symmetric group S₃. Indeed, if B is a finite Boolean ring, then GL₂(B) turns out to be a direct sum of copies of S₃. If B is infinite, then GL₂(B) is more difficult to calculate; we present here descriptions of GL₂(B) for the "extreme" cases of countable Boolean rings—namely, the Boolean ring which is generated by its atoms and the atomless Boolean ring. The former provides a negative answer to the question of whether the functor GL₂(⋅) preserves inverse limits; the latter is a corollary of a theorem which states that, under certain circumstances, GL₂(⋅) preserves direct limits. It turns out, in addition, that every invertible matrix is a product of elementary ones, as is the case for matrices over a Euclidean domain.