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Rings in which one-sided strongly $π$ -regular elements are strongly $π$ -regular

This paper investigates the Hartwig-Luh question regarding the symmetry of one-sided strongly $\pi$ -regularity in Dedekind-finite rings, extending known positive results to new classes while demonstrating that the condition is not left-right symmetric.

Original authors: Dimple Rani Goyal, Dinesh Khurana

Published 2026-01-22

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CC BY 4.0

Original authors: Dimple Rani Goyal, Dinesh Khurana

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, complex city called Ringland. In this city, the buildings are numbers (or abstract objects), and the streets are rules for how you can combine them. The paper you're asking about is a detective story investigating a specific rule in Ringland: Does the order of operations matter when things get stuck?

Here is the breakdown of the paper's story, using simple analogies.

The Big Question: The "One-Way Street" Mystery

In Ringland, there is a special type of building called a Strongly $\pi$ -regular building. Think of this as a building that, if you keep walking through it in a specific direction (multiplying it by itself), you eventually hit a dead end or a loop that lets you turn around and go back.

Right Strongly $\pi$ -regular: You can walk forward, hit a loop, and turn around to go back to the right.
Left Strongly $\pi$ -regular: You can walk forward, hit a loop, and turn around to go back to the left.
Strongly $\pi$ -regular: You can do both. You are perfectly symmetrical; you can turn around no matter which way you face.

The Mystery:
For a long time, mathematicians wondered: If a building allows you to turn around to the right, does it automatically allow you to turn around to the left?

In 1977, two detectives (Hartwig and Luh) asked this question specifically for a type of city called a Dedekind-finite ring. In this city, you can't have a "one-way street" where you can go from A to B and back to A, but not from B to A. It's a city where "A times B equals 1" implies "B times A equals 1."

The Twist:
In 2014, other detectives (Dittmer, Khurana, and Nielsen) found a counter-example. They built a weird city where a building could turn right but not left. So, the answer was "No, not always."

However, they also found that in a specific neighborhood called Exchange Rings, the answer was "Yes."

The Authors' Mission

The authors of this paper (Dimple Rani Goyal and Dinesh Khurana) decided to investigate other neighborhoods in Ringland. They wanted to know:

In which other types of Dedekind-finite cities does the "Right = Left" rule hold?
Can we prove that the "Right = Left" rule is not symmetrical? (i.e., Can a city be "Right-Dischinger" but not "Left-Dischinger"?)

The Good News: Where the Rule Works

The authors found several neighborhoods where the rule holds true. If the city has certain structural properties, then "Right" automatically means "Left."

The "Weakly Semicommutative" Neighborhood: Imagine a neighborhood where if two people bump into each other (multiply to zero), they cause a small, contained mess (nilpotent) that doesn't spread. In this neighborhood, the rule works.
The "One-Sided Duo" Neighborhood: Here, every building is part of a two-way street system. If you can go left, you can go right. The authors proved that if a city is "Left Duo" (everything flows left), it is also "Right Dischinger" (the rule works).
The "Goldie" and "Noetherian" Neighborhoods: These are cities with strict building codes (no infinite chains of smaller and smaller buildings). The authors showed that in these well-ordered cities, the rule holds.
The "Bounded Nilpotence" Neighborhood: If there is a limit to how many times you can multiply a "zero-building" before it stays zero, the rule works.

The Bad News: Where the Rule Fails

The authors then turned their attention to a very specific, weird city they built based on a previous design (by Dittmer et al.). This city is defined by a strange rule: $a = a^2x$ .

They analyzed this city in depth and found it to be a perfect counter-example. It is a city where:

It is Left Dischinger: If you can turn left, you can turn right.
It is NOT Right Dischinger: If you can turn right, you cannot necessarily turn left.

This proves that the property is not left-right symmetric. Just because a city works one way doesn't mean it works the other way.

They also discovered that this weird city has other strange properties:

It is an NR Ring (Nilpotents form a sub-ring), but it still fails the rule.
It is a UU Ring (All "units" or invertible buildings are just "1" plus a "nilpotent" mess), but it still fails the rule.
It is a Linearly McCoy Ring (a specific polynomial rule), but it still fails the rule.

The Conclusion

The paper concludes with a few open questions for future detectives:

We know "Left Duo" implies "Right Dischinger." Does "Left Duo" also imply "Left Dischinger"? (We don't know yet).
Does being a "McCoy Ring" (a polynomial rule) guarantee the Dischinger property? (We don't know yet).

Summary in One Sentence

The authors mapped out the landscape of Ringland, proving that in some well-behaved neighborhoods, "turning right" guarantees "turning left," but they also constructed a bizarre, asymmetric city where you can turn right but get stuck when trying to turn left, proving that these two directions are not always the same.

Technical Summary: Rings in which One-Sided Strongly $\pi$ -Regular Elements are Strongly $\pi$ -Regular

Problem Statement
The paper addresses a fundamental question in ring theory regarding the symmetry of regularity conditions. Specifically, it investigates the condition under which a one-sided strongly $\pi$ -regular element is necessarily strongly $\pi$ -regular (i.e., both left and right strongly $\pi$ -regular).

This inquiry stems from a 1977 question posed by Hartwig and Luh: If $S$ is a Dedekind-finite ring and $a \in S$ satisfies $aS = a^2S$ , does it follow that $Sa = Sa^2$ ? While Dittmer, Khurana, and Nielsen (2014) provided a negative answer for general Dedekind-finite rings, they proved the statement holds for Dedekind-finite exchange rings. The authors aim to explore this question for other classes of Dedekind-finite rings and determine if the property of being a "Dischinger ring" (where one-sided strong $\pi$ -regularity implies strong $\pi$ -regularity) is left-right symmetric.

Methodology and Definitions
The authors utilize the framework of strongly $\pi$ -regular elements, defined as elements $a$ where $a^n = a^{n+1}x$ for some $n \ge 1$ and $x \in R$ . A ring is termed right Dischinger if every right strongly $\pi$ -regular element is strongly $\pi$ -regular.

The methodology involves:

Theoretical Derivation: Establishing equivalences between one-sided regularity conditions and properties of units, idempotents, and annihilators.
Counter-Example Construction: Analyzing the specific ring $R = F\langle a, x : a = a^2x \rangle$ (where $F$ is a commutative ring with trivial idempotents), originally constructed by Dittmer et al. to show that Dedekind-finite rings need not be right Dischinger.
Structural Analysis: The authors perform a detailed analysis of the monomial basis of $R$ (using a specific total order $\prec$ on reduced monomials) to characterize its nilpotent elements, zero divisors, and units. This allows them to test various ring properties against the Dischinger condition.

Key Contributions and Results

Generalization of Azumaya's Theorem: The authors generalize Azumaya's result that rings with bounded nilpotence index are Dischinger. They prove (Theorem 2.10) that a ring with identity is right Dischinger if it satisfies specific chain conditions on principal right ideals or left annihilators involving nilpotent elements.
Sufficient Conditions for Dischinger Rings:
- Weakly Semicommutative Rings: Proven to be Dischinger (Corollary 2.5).
- NI and NR Rings: While NI rings (where nilpotents form an ideal) are Dischinger, the authors show that NR rings (where nilpotents form a subring) are not necessarily right Dischinger.
- One-Sided Duo Rings: Weakly left duo rings with identity are proven to be right Dischinger (Corollary 2.8).
- Left Goldie Rings: Proven to be Dischinger (Corollary 2.14), extending the result to rings with finite uniform dimension and ACC on left annihilators.
Left-Right Asymmetry: The central contribution is the proof that the Dischinger condition is not left-right symmetric.
- The authors demonstrate that the ring $R = F_2\langle a, x : a = a^2x \rangle$ is left Dischinger but not right Dischinger (Theorem 3.10).
- They establish that in this ring, the set of left zero divisors is $Ra$, while the set of right zero divisors is $\bigcup r_R(a^n)$ .
- They prove that $R$ satisfies the ascending chain condition (ACC) on left annihilators but fails to be right Dischinger, showing that ACC on left annihilators alone is insufficient for the right Dischinger property.
Characterization of Specific Classes:
- UU Rings: The ring $R$ is shown to be a UU ring (all units are unipotent), yet it is not right Dischinger.
- McCoy Rings: The ring $R$ is proven to be both left and right linearly McCoy, yet it is not right Dischinger. This leaves open the question of whether McCoy rings with identity are Dischinger.

Significance and Claims
The paper claims to resolve the local validity of Dischinger's result for various classes of Dedekind-finite rings. By rigorously analyzing the ring $R$ , the authors provide a definitive counter-example showing that:

A ring can be left Dischinger without being right Dischinger.
Properties such as being an NR ring, a UU ring, or satisfying the ACC on left annihilators do not guarantee the right Dischinger property.
The condition "one-sided strongly $\pi$ -regular implies strongly $\pi$ -regular" is inherently asymmetric in general rings.

The authors conclude by posing several open questions, including whether right or left McCoy rings with identity are Dischinger, and whether weakly left duo rings are left Dischinger, indicating that the classification of Dischinger rings remains an active area of investigation.