[Paper Review] Strongly J-Clean Rings with Involutions

Huanyin Chen, Abdullah Harmancı+1 more|arXiv (Cornell University)|Jul 2, 2012

Rings, Modules, and Algebras16 references3 citations

TL;DR

This paper introduces and characterizes strongly $J$-clean rings with involutions, proving that such rings are equivalent to uniquely strongly $*$-clean rings—where each element uniquely decomposes into a projection and an invertible element commuting with it. The key contribution is establishing that strong $J$-cleanness with involution is equivalent to unique strong $*$-cleanness, resolving a structural gap in the non-involutive case and unifying key ring-theoretic properties under involution.

ABSTRACT

A ring with an involution * is called strongly $J$-*-clean if every element is a sum of a projection and an element of the Jacobson radical that commute. In this article, we prove several results characterizing this class of rings. It is shown that a *-ring $R$ is strongly $J$-*-clean, if and only if $R$ is uniquely clean and strongly *-clean, if and only if $R$ is uniquely strongly *-clean, that is, for any $a\in R$, there exists a unique projection $e\in R$ such that $a-e$ is invertible and $ae=ea$.

Motivation & Objective

• To define and study strongly $J$-clean rings equipped with an involution, extending the theory of strongly clean and $J$-clean rings to the $*$-ring setting.
• To resolve the structural gap between strongly $J$-clean and uniquely strongly $*$-clean rings by proving their equivalence under involution.
• To characterize strongly $J$-clean $*$-rings via conditions on the Jacobson radical and uniqueness of projections.
• To explore preservation of strong $J$-cleanness under constructions like power series rings, group rings, and polynomial quotients.

Proposed method

• Introduce the concept of strongly $J$-$*$-clean rings: every element is a sum of a projection and an element in the Jacobson radical, with commuting components.
• Use the equivalence of strongly $J$-clean and strongly $*$-clean rings to derive structural conditions under involution.
• Apply the uniqueness of projections in the decomposition to show equivalence with uniquely strongly $*$-clean rings.
• Leverage the fact that $2 \in J(R)$ and that idempotents are projections in abelian $*$-rings to derive radical and commutativity conditions.
• Use lifting properties modulo the Jacobson radical to construct decompositions in power series and group rings.
• Prove preservation of strong $J$-cleanness under $R[[x]]$, $R[[x]]/(x^n)$, and group rings $RG$ for locally finite $2$-groups.

Experimental results

Research questions

• RQ1 Is there a characterization of strongly $J$-clean $*$-rings that unifies strong $*$-cleanness and unique projection decomposition?
• RQ2 Does the presence of an involution force strong $J$-cleanness to coincide with unique strong $*$-cleanness, unlike the non-involutive case?
• RQ3 Under what conditions on $R$ and $G$ is the group ring $RG$ strongly $J$-$*$-clean?
• RQ4 How does the Jacobson radical interact with the involution in strongly $J$-$*$-clean rings?
• RQ5 What ring constructions preserve strong $J$-$*$-cleanness, and what are the necessary and sufficient conditions?

Key findings

• A $*$-ring $R$ is strongly $J$-$*$-clean if and only if it is uniquely strongly $*$-clean, establishing a direct equivalence not present in the non-involutive case.
• Strongly $J$-$*$-clean rings are abelian, with all idempotents being projections and $2 \in J(R)$.
• The Jacobson radical satisfies $J(R) = \{x \in R \mid 1 - x \text{ is invertible}\}$, characterizing the radical in terms of invertibility.
• The power series ring $R[[x]]$ is strongly $J$-$*$-clean if and only if $R$ is strongly $J$-$*$-clean.
• The group ring $RG$ is strongly $J$-$*$-clean if and only if $R$ is strongly $J$-$*$-clean and $G$ is a $2$-group.
• The ring $R[x]/(x^2)$ is strongly $J$-$*$-clean if and only if $R$ is strongly $J$-$*$-clean, showing preservation under quadratic polynomial quotients.