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Centers in domains with quadratic growth

Agata Smoktunowicz (2005)

Open Mathematics

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.

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 ∈ 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 ℤ 3 , B or ℤ 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 .

Rings consisting entirely of certain elements

Huanyin Chen, Marjan Sheibani, Nahid Ashrafi (2018)

Czechoslovak Mathematical Journal

We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ℤ 3 ⊕ ℤ 3 ; ℤ 3 ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M 2 ( ℤ 2 ) or M 2 ( ℤ 3 ) ; or the ring of a Morita context with zero pairings where the underlying rings are ℤ 2 or ℤ 3 .

Single elements.

Gardner, B.J., Mason, Gordon (2006)

Beiträge zur Algebra und Geometrie

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