[Paper Review] Rings over which every matrix is the sum of a tripotent and a nilpotent

Marjan Sheibani, H Chen|arXiv (Cornell University)|Feb 18, 2017

Rings, Modules, and Algebras9 references3 citations

TL;DR

This paper establishes that matrix rings over 2-primal strongly 2-nil-clean rings and over strongly 2-nil-clean rings of bounded index are trinil clean, meaning every matrix is expressible as the sum of a tripotent and a nilpotent matrix. The key contribution is a structural characterization that extends known results on nil-clean and weakly nil-clean rings to a broader class of rings via tripotent decomposition.

ABSTRACT

A ring $R$ is trinil clean if every element in $R$ is the sum of a tripotent and a nilpotent. If $R$ is a 2-primal strongly 2-nil-clean ring, we prove that $M_n(R)$ is trinil clean for all $n\in {\Bbb N}$. Furthermore, we show that the matrix ring over a strongly 2-nil-clean ring of bounded index is trinil clean. We thereby provide various type of rings over which every matrix is the sum of a tripotent and a nilpotent.

Motivation & Objective

• To investigate when matrix rings over a ring are trinil clean, i.e., every matrix is the sum of a tripotent and a nilpotent matrix.
• To extend the theory of nil-clean and weakly nil-clean rings by introducing trinil clean rings, defined by decomposition into tripotent and nilpotent elements.
• To characterize conditions under which matrix rings over strongly 2-nil-clean rings are trinil clean, particularly in the context of 2-primal and bounded index rings.
• To provide sufficient conditions ensuring that matrix rings inherit the trinil clean property from the base ring, generalizing prior results on nil-clean matrix rings.

Proposed method

• Use of the Chinese Remainder Theorem to decompose a trinil clean ring R into R ≅ A × B, where 2 ∈ J(A) and 3 ∈ J(B), based on the fact that 6 ∈ N(R).
• Leverage the equivalence between strongly 2-nil-clean rings and rings where a − a³ ∈ N(R) for all a ∈ R, enabling structural decomposition and transfer of properties.
• Apply Theorem 2.7, which states that a ring R is trinil clean iff R/J(R) is trinil clean and J(R) is nil, to lift the trinil clean property from quotient rings to the original ring.
• Use the fact that matrix rings over nil-clean rings are nil-clean (via known results) and that matrix rings over quotient rings of strongly 2-nil-clean rings inherit trinil clean structure via quotient isomorphisms.
• Employ Lemma 4.8, which states that if R is of bounded index and J(R) is nil, then Mₙ(R) is nil, to establish nilpotency in matrix rings.
• Apply Corollary 3.3 to show that if R is strongly 2-nil-clean and of bounded index, then J(R) is nil and R/J(R) is tripotent, enabling inductive lifting of the trinil clean property to matrix rings.

Experimental results

Research questions

• RQ1 Under what conditions on a ring R is the matrix ring Mₙ(R) trinil clean, i.e., every matrix is the sum of a tripotent and a nilpotent matrix?
• RQ2 How do the properties of 2-primal rings and strongly 2-nil-clean rings interact to ensure that Mₙ(R) is trinil clean for all n ∈ ℕ?
• RQ3 What role does the bounded index condition play in ensuring that matrix rings over strongly 2-nil-clean rings are trinil clean?
• RQ4 Can the trinil clean property of a ring R be lifted to Mₙ(R) when R is a finite product of rings with specific Jacobson radical and quotient structure?
• RQ5 To what extent do 2-Boolean rings and weakly nil-clean rings satisfy the trinil clean condition in their matrix rings?

Key findings

• If R is a 2-primal strongly 2-nil-clean ring, then Mₙ(R) is trinil clean for all n ∈ ℕ, extending known results on nil-clean matrix rings.
• Matrix rings over strongly 2-nil-clean rings of bounded index are trinil clean, providing a broad class of examples where every matrix decomposes into a tripotent and a nilpotent.
• The ring ℤₘ is trinil clean if and only if m = 2ᵏ3ˡ with k, l ∈ ℕ⁺ and k + l ≠ 0, and in such cases, Mₙ(ℤₘ) is trinil clean for all n.
• For any ring R where (a − a³)ᵐ = 0 for all a ∈ R and some fixed m ∈ ℕ, the matrix ring Mₙ(R) is trinil clean for all n ∈ ℕ.
• Every 2-Boolean ring R satisfies (a − a³)³ = 0 for all a ∈ R, so Mₙ(R) is trinil clean for all n ∈ ℕ, as shown via Corollary 4.10.
• The Jacobson radical J(R) of a trinil clean ring is always nil, and R/J(R) is trinil clean, which enables lifting the trinil clean structure to matrix rings via quotient and ideal-theoretic techniques.