From Topologies of a Set to Subrings of Its Power Set

Jaballah, Ali; Jarboui, Noômen

doi:10.1017/s0004972720000015

Cited by 3 publications

(15 citation statements)

References 13 publications

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“…In contrast to the setting of the preceding paragraph, [19] studied rings R that are exquisitely commutative, specifically, rings of the form R = P(S) for some (usually nonempty, often finite) set S. As is well known (and easy to see), P(S) is a Boolean ring, with symmetric difference playing the role of addition and intersection playing the role of multiplication, so that S is the multiplicative identity element of P(S). If |S| ≥ 2, we will identify {∅, S} with F 2 , so that F 2 ⊆ P(S).…”

Section: Introductionmentioning

confidence: 92%

“…If |S| ≥ 2, we will identify {∅, S} with F 2 , so that F 2 ⊆ P(S). We next discuss the first of the results from [19] that we are generalizing here. Apart from a topologically formulated equivalent condition that we will mention later, [19,Theorem 2.3] can be stated as follows.…”

Section: Introductionmentioning

confidence: 95%

“…We next discuss the first of the results from [19] that we are generalizing here. Apart from a topologically formulated equivalent condition that we will mention later, [19,Theorem 2.3] can be stated as follows. Let S be a nonempty set and R := P(S).…”

Section: Introductionmentioning

confidence: 95%

“…The first goal of this work is to generalize some recent results from [19] to a noncommutative setting and thereby deepen our understanding of finite Boolean rings. Accomplishing the first goal will involve extending the context for the FIP, FCP and FMC concepts to noncommutative ring extensions.…”

Section: Introductionmentioning

confidence: 99%

“…The next paragraph announces some conventions and notation. After that, four paragraphs provide some relevant background, then one paragraph states the results from [19] that we are generalizing, then one paragraph gives details about our first main result (Theorem 2.2) and explains some ways in which it generalizes the above-mentioned results from [19], and then two paragraphs describe our results for arbitrary prime characteristic p, including a catenarian conclusion (Corollary 2.10 (b)) that strengthens an inequality that had been observed in [19] in the Boolean context.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Characterizing finite Boolean rings by using finite chains of subrings

Jarboui

Dobbs

2021

GJOM

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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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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 95%