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Certain decompositions of matrices over Abelian rings

Nahid Ashrafi, Marjan Sheibani, Huanyin Chen (2017)

Czechoslovak Mathematical Journal

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n . We prove that M n ( R ) is nil clean if and only if R / J ( R ) is Boolean and M n ( J ( R ) ) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R / J ( R ) is 3 , B or 3 B where B is a Boolean ring, and that M n ( R ) is weakly nil clean if and only if M n ( R ) is nil clean for all n 2 .

Certain partitions on a set and their applications to different classes of graded algebras

Antonio J. Calderón Martín, Boubacar Dieme (2021)

Communications in Mathematics

Let ( 𝔄 , ϵ u ) and ( 𝔅 , ϵ b ) be two pointed sets. Given a family of three maps = { f 1 : 𝔄 𝔄 ; f 2 : 𝔄 × 𝔄 𝔄 ; f 3 : 𝔄 × 𝔄 𝔅 } , this family provides an adequate decomposition of 𝔄 { ϵ u } as the orthogonal disjoint union of well-described -invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak H * -algebras.

Certain simple maximal subfields in division rings

Mehdi Aaghabali, Mai Hoang Bien (2019)

Czechoslovak Mathematical Journal

Let D be a division ring finite dimensional over its center F . The goal of this paper is to prove that for any positive integer n there exists a D ( n ) , the n th multiplicative derived subgroup such that F ( a ) is a maximal subfield of D . We also show that a single depth- n iterated additive commutator would generate a maximal subfield of D .

Characterization of automorphisms of Radford's biproduct of Hopf group-coalgebra

Xing Wang, Daowei Lu, Ding-Guo Wang (2024)

Czechoslovak Mathematical Journal

We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford’s π -biproduct. Firstly, we discuss the endomorphism monoid End π -Hopf ( A × H , p ) and the automorphism group Aut π -Hopf ( A × H , p ) of Radford’s π -biproduct A × H = { A × H α } α π , and prove that the automorphism has a factorization closely related to the factors A and H = { H α } α π . What’s more interesting is that a pair of maps ( F L , F R ) can be used to describe a family of mappings F = { F α } α π . Secondly, we consider the relationship between the automorphism group Aut π -Hopf ( A × H , p ) and the automorphism group Aut π - 𝒴 𝒟 -Hopf ( A ) of A , and...

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