[Paper Review] On strongly $g(x)$-clean rings

Lingling Fan, Xiande Yang|ArXiv.org|Mar 24, 2008

Rings, Modules, and Algebras13 references3 citations

TL;DR

This paper introduces and investigates strongly $g(x)$-clean rings, a generalization of strongly clean rings where each element decomposes into a sum of a root of $g(x)$ and a unit that commute. It establishes that strong $g(x)$-cleanness and strong cleanness are equivalent when $g(x) = (x-a)(x-b)$ and $b-a$ is a unit, and characterizes rings generated by units that are strongly $g(x)$-clean for specific polynomials like $x^2 - nx$. The key contribution is a unified framework linking polynomial roots, unit decomposition, and ring structure in noncommutative rings.

ABSTRACT

Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a polynomial in the polynomial ring $C(R)[x]$. $R$ is called strongly $g(x)$-clean if every element $r \in R$ can be written as $r=s+u$ with $g(s)=0$, $u$ a unit of $R$, and $su=us$. The relation between strongly $g(x)$-clean rings and strongly clean rings is determined, some general properties of strongly $g(x)$-clean rings are given, and strongly $g(x)$-clean rings generated by units are discussed.

Motivation & Objective

• To define and study strongly $g(x)$-clean rings as a generalization of strongly clean rings.
• To determine the precise conditions under which strong $g(x)$-cleanness implies strong cleanness and vice versa.
• To characterize classes of rings generated by units that are strongly $g(x)$-clean for specific polynomials such as $x^2 - nx$.
• To explore structural properties of strongly $g(x)$-clean rings, especially in relation to idempotents and roots of unity.

Proposed method

• Define strongly $g(x)$-clean rings via decomposition $r = s + u$, where $g(s) = 0$, $u$ is a unit, and $s$ commutes with $u$.
• Use ring homomorphisms and epimorphisms to transfer the $g(x)$-cleanness property across rings.
• Apply the theory of idempotents and unit decomposition via Lemma 4.6 to analyze elements in $R$.
• Establish equivalence between strong $g(x)$-cleanness and strong cleanness for quadratic polynomials $g(x) = (x-a)(x-b)$ when $b-a$ is a unit.
• Analyze matrix rings over $C(X)$, $C^*(X)$, and local rings to demonstrate strong $g(x)$-cleanness for $g(x) = x^2 - nx$.
• Prove symmetry in strong cleanness for $g(x) = ax^{2n} \pm bx$ by showing equivalence between $-x$ substitution and unit closure.

Experimental results

Research questions

• RQ1 Under what conditions is a strongly $g(x)$-clean ring also strongly clean, and vice versa?
• RQ2 When does strong $g(x)$-cleanness imply strong cleanness for quadratic polynomials $g(x) = (x-a)(x-b)$?
• RQ3 Which rings generated by units are strongly $g(x)$-clean for $g(x) = x^2 - nx$?
• RQ4 How do structural properties like idempotents and roots of unity relate to strong $g(x)$-cleanness in matrix rings?
• RQ5 Is strong $(x^{2n+1} - x)$-cleanness equivalent to strong $(x^{2n+1} + x)$-cleanness?

Key findings

• Strong $g(x)$-cleanness and strong cleanness are equivalent for $g(x) = (x-a)(x-b)$ if and only if $b-a$ is a unit in $R$, establishing a precise algebraic condition for equivalence.
• The ring $\mathbb{Z}_{(7)}C_3$ is strongly $(x^6 - 1)$-clean and strongly $(x^4 - x)$-clean but not strongly clean, showing that strong $g(x)$-cleanness does not imply strong cleanness.
• For any $n \in \mathbb{N}$, if $X$ is strongly zero-dimensional, then $C(X)$ and $C^*(X)$ are strongly $(x^2 - nx)$-clean, and if $X$ is a $P$-space, then $\mathbb{M}_k(C(X))$ is strongly $(x^2 - nx)$-clean.
• Matrix rings over locally Artinian rings, such as $\mathbb{M}_k(E)$ with $E = \mathrm{End}_F(V)$, are strongly $(x^2 - nx)$-clean when $\mathrm{char}(F) \nmid n$, and every element is a sum of a unit and a square root of 1 that commutes.
• If $R$ is strongly $(x^2 + x + 1)$-clean, then $R = U_2(R)$ and $R$ is strongly $(x^4 - x)$-clean, with every element expressible as a sum of a unit and a cubic root of 1 that commutes.
• For $n \geq 2$, in a strongly $(x^n - x)$-clean ring, every element is either a sum of a unit and a $(n-1)$-st root of unity that commutes, or both left and right ideals contain non-trivial idempotents.