# Characterizing finite Boolean rings by using finite chains of subrings

## Main Article Content

## 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, *R*_{0}:=F_{2} ⊂ ... ⊂ *R*_{n}=*R*, going from F_{2} 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, ℛ_{0}:=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.

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## Article Details

*Gulf Journal of Mathematics*,

*10*(1), 69-94. https://doi.org/10.56947/gjom.v10i1.557