Characterizing finite Boolean rings by using finite chains of subrings

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Noomen Jarboui
David Dobbs

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 ⊂ ... ⊂ Rn=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 Bn, the nth 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 Fp 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:=Fp ⊂ ... ⊂ ℛm=R, going from Fp to R, such that ℛm-1 is a commutative ring and R is a unital ℛm-1-algebra. Additional characterizations, applications, examples and remarks are given.

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How to Cite
Jarboui, N., & Dobbs, D. (2021). Characterizing finite Boolean rings by using finite chains of subrings. Gulf Journal of Mathematics, 10(1), 69-94. Retrieved from https://gjom.org/index.php/gjom/article/view/557
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