Characterizing finite Boolean rings by using finite chains of subrings

Abstract

Let R be a nonzero associative ring with identity. It is proved that R is a finite Boolean ring if (and only if) 1 is the only unit of R and there exists a finite maximal chain C each of whose n steps is a proper unital ring extension, R₀:=F₂ ⊂ ... ⊂ R_n=R, going from F₂ to R. If these equivalent conditions hold and R has exactly n maximal ideals, then any such C has length n-1 and the number of unital subrings of R is B_n, the n^th Bell number. It is also proved that if R has characteristic p for some prime number p, then R is isomorphic to a finite direct product of copies of F_p if (and only if) for some integer m ≥ 0, R has exactly m+1 maximal ideals and there exists a finite maximal chain of proper unital ring extensions, ℛ₀:=F_p ⊂ ... ⊂ ℛ_m=R, going from F_p to R, such that ℛ_m-1 is a commutative ring and R is a unital ℛ_m-1-algebra. Additional characterizations, applications, examples and remarks are given.