Maximal Subrings and Covering Numbers of Finite Semisimple Rings

Peruginelli, Giulio; Werner, Nicholas J.

doi:10.1080/00927872.2018.1455099

Cited by 8 publications

(12 citation statements)

References 25 publications

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“…Thus, σ(R) = min{σ(S), σ(S 1 ), σ(S 2 )}. Since S 2 is not coverable and σ(S) = σ(S 1 ) by [27,Proposition 5.4], we get σ(R) = σ(S 1 ) = σ(M n (q 1 )). Finally, when n = 2, n − (n/a) = 1, so d n − (n/a) and R is not σ-elementary.…”

Section: Frequently Used Resultsmentioning

confidence: 99%

“…Following Proposition 3.3, we seek a minimal cover by maximal subrings of S of the union of the centralizers C S (x). By [27,Theorem 4.5], any maximal subring of S has the form M 1 ⊕ S 2 or S 1 ⊕ M 2 , where M i is a maximal subring of S i . If q 2 > p, then the maximal subrings of S 2 ∼ = F q 2 are the maximal subfields of F q 2 ; otherwise, {0} is the only maximal subring of S 2 .…”

Section: 2mentioning

confidence: 99%

“…Indeed, Kappe [18, p. 87] further writes, "An interesting question would be if there are integers n > 2 that are not the covering number of a ring." Other recent works on this problem include [5,9,27,32,34].…”

Section: Introductionmentioning

confidence: 99%

“…The formula for the covering number of M n (q) is due to Crestani, Lucchini, and Maróti [9,23]. Since M n (q) is simple, it is clearly σ-elementary, and the fact that these are the only noncommutative semisimple σ-elementary rings follows from the classification of covering numbers for finite semisimple rings done in [27]. The contributions of the present paper include the recognition that the rings A(n, q 1 , q 2 ) may be σ-elementary, the determination of the covering number of A(n, q 1 , q 2 ), and a proof that any σ-elementary ring falls into one of the four classes listed in Theorem 1.3.…”

Section: Introductionmentioning

confidence: 99%

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The covering numbers of rings

Swartz¹,

Werner²

2021

Preprint

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A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ(R) of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if σ(R) < σ(R/I) for every nonzero two-sided ideal I of R. If R is a ring with unity, then we define the unital covering number σ u (R) to be the size of a minimal cover of R by subrings that contain 1 R (if such a cover exists), and R is σ u -elementary if σ u (R) < σ u (R/I) for every nonzero two-sided ideal of R. In this paper, we classify all σ-elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all σ u -elementary rings and prove σ u (R) = σ(R) for every σ u -elementary ring R. We also prove that, if R is a ring without unity with a finite cover, then there exists a unital ring R ′ such that σ(R) = σ u (R ′ ), which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, if E (N ) := {m : m N, σ(R) = m for some ring R}, then we show that |E (N )| = Θ(N/ log(N )), which proves that almost all integers are not covering numbers of a ring.

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Section: Frequently Used Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The covering numbers of rings

Swartz¹,

Werner²

2021

Preprint

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show abstract

“…It is natural to try and extend such results to other settings. For instance, coverings of rings have been studied (including very recently) in [7,9,13,24,28,39]. Yet another motivation comes from the short note [22], where the first named author found a sharp bound for the number of proper subspaces of a fixed codimension d !…”

Section: Introductionmentioning

confidence: 99%

Covering modules by proper submodules

Khare

Tikaradze

2021

Communications in Algebra

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A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. In this note we cover R-modules by proper submodules for commutative rings R, thereby subsuming and recovering both cases above. Specifically, we study the smallest cardinal number @, possibly infinite, such that a given R-module is a union of @-many proper submodules. (1) We completely characterize when @ is a finite cardinal; this parallels for modules a 1954 result of Neumann. ( 2) We also compute the covering (cardinal) numbers of finitely generated modules over quasi-local rings and PIDs, recovering past results for vector spaces and abelian groups respectively. (3) As a variant, we compute the covering number of an arbitrary direct sum of cyclic monoids. Our proofs are self-contained.

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The covering numbers of rings

Swartz,

Werner

2024

Journal of Algebra

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Maximal Subrings and Covering Numbers of Finite Semisimple Rings

Cited by 8 publications

References 25 publications

The covering numbers of rings

The covering numbers of rings

Covering modules by proper submodules

The covering numbers of rings

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