Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1217-1226, 2022


Certain additive decompositions in a noncommutative ring

Huanyin Chen, Marjan Sheibani, Rahman Bahmani

Received January 30, 2022.   Published online July 21, 2022.

Abstract:  We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J (M_2(R))$, or $I_2-A\in J(M_2(R))$, or $A$ is similar to $\left(\smallmatrix0&\lambda
1&\mu\right)$, where $\lambda\in J(R)$, $\mu\in1+J(R)$, and the equation $x^2-x\mu-\lambda=0$ has a root in $J(R)$ and a root in $1+J(R)$. We further prove that $f(x)\in R[[x]]$ is strongly $J$-clean if $f(0)\in R$ be optimally $J$-clean.
Keywords:  idempotent matrix; nilpotent matrix; projective-free ring; quadratic equation; power series
Classification MSC:  15A09, 16E50, 16U60
DOI:  10.21136/CMJ.2022.0039-22


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Affiliations:   Huanyin Chen, School of Mathematics, Hangzhou Normal University, No. 2318, Yuhangtang Road, 311121, Hangzhou, P. R. China, e-mail: huanyinchen@aliyun.com; Marjan Sheibani (corresponding author), Farzanegan Campus, Semnan University, 17 Shahrivar Blvd. Semnan, Iran, e-mail: m.sheibani@semnan.ac.ir; Rahman Bahmani, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195-363, Semnan, Iran, e-mail: rbahmani@semnan.ac.ir


 
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