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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 ∈ 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 .

Commutative group algebras of highly torsion-complete abelian p -groups

Peter Vassilev Danchev (2003)

Commentationes Mathematicae Universitatis Carolinae

A new class of abelian p -groups with all high subgroups isomorphic is defined. Commutative modular and semisimple group algebras over such groups are examined. The results obtained continue our recent statements published in Comment. Math. Univ. Carolinae (2002).

Commutative modular group algebras of p -mixed and p -splitting abelian Σ -groups

Peter Vassilev Danchev (2002)

Commentationes Mathematicae Universitatis Carolinae

Let G be a p -mixed abelian group and R is a commutative perfect integral domain of char R = p > 0 . Then, the first main result is that the group of all normalized invertible elements V ( R G ) is a Σ -group if and only if G is a Σ -group. In particular, the second central result is that if G is a Σ -group, the R -algebras isomorphism R A ≅ R G between the group algebras R A and R G for an arbitrary but fixed group A implies A is a p -mixed abelian Σ -group and even more that the high subgroups of A and G are isomorphic, namely, ℋ A ≅ ℋ G . Besides,...

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