Commutators for near-rings: Huq $${\neq}$$ ≠ Smith

Janelidze, George; Márki, László; Veldsman, Stefan

doi:10.1007/s00012-016-0398-8

Cited by 6 publications

(2 citation statements)

References 14 publications

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“…The list of known algebraic structures whose categories have the two commutators different from each other includes loops (where, as mentioned in [28], this can be deduced from the result of Exercise 10 of Chapter 5 in [24]), digroups (=sets equipped with two independent group structures with the same identity element; see e.g. [13]), and near-rings (as shown in [40]; note, on the other hand, that prime ideals of near-rings were introduced and studied in [64]). (f) There is more to say about various non-modular commutators introduced in universal algebra, the relative commutator in the sense of T. Everaert and T. Van der Linden [21], and commutator theory and related studies in regular (not necessarily Barr exact) categories developed in several papers of D. Bourn and M. Gran, but we omit it here.…”

Section: Ideals In An Abstract Commutative Worldmentioning

confidence: 99%

Abstractly constructed prime spectra

Facchini¹,

Finocchiaro²,

Janelidze³

2021

Preprint

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The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum Spec(R) of a unital commutative ring R is always a spectral (=coherent) topological space. In this generalization, which includes several other known ones, the role of ideals of R is played by elements of an abstract complete lattice L equipped with binary multiplication with xy x ∧ y for all x, y ∈ L. In fact when no further conditions on L are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral. This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of L; we also make short additional remarks on semiprime elements. We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world. The cases of groups and of non-commutative rings are briefly considered separately.

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Section: Ideals In An Abstract Commutative Worldmentioning

confidence: 99%

Abstractly constructed prime spectra

Facchini¹,

Finocchiaro²,

Janelidze³

2021

Preprint

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“…We show that Huq=Smith for left skew braces. Notice that Huq = Smith for digroups and near-rings [18]. We give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer.…”

Section: Introductionmentioning

confidence: 99%

Aspects of the Category SKB of Skew Braces

Bourn¹,

Facchini²,

Mara³

2022

Preprint

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We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, "product" of ideals of skew braces is often considered. We show that Huq=Smith for left skew braces. Finally, we give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer.

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On when the union of two algebraic sets is algebraic

Aichinger,

Behrisch,

Rossi

2024

Aequat. Math.

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In universal algebraic geometry, an algebra is called an equational domain if the union of two algebraic sets is algebraic. We characterize equational domains, with respect to polynomial equations, inside congruence permutable varieties, and with respect to term equations, among all algebras of size two and all algebras of size three with a cyclic automorphism. Furthermore, for each size at least three, we prove that, modulo term equivalence, there is a continuum of equational domains of that size.

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Commutators for near-rings: Huq $${\neq}$$ ≠ Smith

Abstract: Abstract. It is shown that the Huq and the Smith commutators do not coincide in the variety of near-rings.

Cited by 6 publications

References 14 publications

Abstractly constructed prime spectra

Abstractly constructed prime spectra

Aspects of the Category SKB of Skew Braces

On when the union of two algebraic sets is algebraic

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