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Extensions of Centrally Essential Rings

This paper characterizes almost fully prime centrally essential rings by examining their structure through ideal extensions, centrally essential Dorroh extensions, and trivial extensions.

Original authors: Oleg Lyubimtsev, Askar Tuganbaev

Published 2026-06-04

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CC BY 4.0

Original authors: Oleg Lyubimtsev, Askar Tuganbaev

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, bustling city called Ring City. In this city, every building (an element of the ring) has a specific job. Usually, in a "normal" city (a commutative ring), everyone gets along perfectly; it doesn't matter who you talk to first, the conversation goes the same way ( $A \times B = B \times A$ ).

But in the special city described in this paper, called a Centrally Essential Ring, things are a bit more chaotic. Here, some buildings are "rebellious" and don't follow the standard rules of order. However, there is a strict safety rule: No rebel is allowed to be truly alone.

The Golden Rule: "Find a Friend in the Center"

The paper defines a Centrally Essential Ring with a simple, almost poetic rule:

If you pick any non-zero building (element) in the city, no matter how rebellious it is, you can always find a "peacekeeper" (a central element) to talk to, such that when they interact, the result is also a peacekeeper.

Think of it like a chaotic party. If you grab a rowdy guest (element $a$ ), you can always find a calm host (central element $x$ ) to shake their hand. When they shake hands, the result is another calm host (central element $y$ ). Even the most chaotic guest must have a connection to the calm center.

The Main Discovery: The "Almost Perfect" City

The authors, Oleg and Askar, focused on a specific type of city called an "Almost Fully Prime" ring. Imagine a city where almost every neighborhood (ideal) is perfectly organized and indivisible.

They discovered that for these specific cities to be "Centrally Essential" (where the Golden Rule holds), the city must have a very specific, almost boring structure:

One Tiny Secret: The city has exactly one tiny, hidden neighborhood (a unique non-zero proper ideal) that sits right in the middle of everything.
The Outer World is a Field: If you ignore that tiny secret neighborhood, the rest of the city looks exactly like a Field (a mathematical world where you can divide everything perfectly, like the numbers you learn in school).
The Connection: The entire city is essentially built by taking that "Field" and attaching that one tiny secret neighborhood to it.

The Analogy: Imagine a perfect, transparent glass sphere (the Field). Now, imagine gluing a single, tiny, opaque speck of dust (the central ideal) right in the center. The paper says: If your ring is "Almost Fully Prime" and "Centrally Essential," it must look exactly like that glass sphere with one speck of dust. If you have two specks, or if the speck isn't in the center, the magic breaks.

Building New Cities: The "Extensions"

The paper also explains how to build these special cities using two construction methods:

1. The Dorroh Extension (The "Add-on" Method)
Imagine you have a base city $R$ and a new material $S$ . You can build a bigger city by stacking them together. The paper proves that if your base city is "Centrally Essential," the new stacked city will also be "Centrally Essential," and vice versa. It's like saying: "If the foundation is stable, the whole house is stable."

2. The Trivial Extension (The "Ghost Layer" Method)
This is a specific way of building where the new material $M$ doesn't interact with itself (it's "ghostly"). You layer it on top of the ring $R$ .

The paper gives a checklist (Theorem 1.3) to see if this new layered city works.
The Check: You have to verify that every piece of the ghost layer can find a "peacekeeper" in the base ring, and every piece of the base ring can find a "peacekeeper" in the ghost layer. If everyone can find a friend in the other layer, the whole new city is "Centrally Essential."

The "Derivation" Twist

In one of their examples, the authors show how to build a non-commutative (chaotic) city using derivations. Think of a derivation as a "rule of change" (like a derivative in calculus).

They take a field (a perfect world) and apply two different, conflicting rules of change.
These rules are "incomparable," meaning they don't agree on anything.
By arranging these rules into a matrix (a grid of numbers), they create a city that is chaotic (non-commutative) but still follows the Golden Rule. It's like a dance where two dancers move to different beats, yet somehow, they never lose their connection to the music's center.

Summary in Plain English

This paper is a map for a very specific type of mathematical universe. It tells us:

What it looks like: A perfect world with exactly one tiny, central imperfection.
How to build it: You can create these worlds by stacking materials together, but you have to check that the "peacekeepers" (central elements) can reach everyone.
Why it matters: It helps mathematicians understand the boundary between "perfectly ordered" worlds and "chaotic" worlds. It shows that even in a chaotic, non-commutative ring, there is a hidden order if you look for the "central" connections.

The authors didn't just say "it exists"; they gave us the exact blueprint (Theorem 1.2) and the construction manual (Theorem 1.3) to build and recognize these unique mathematical structures.

Technical Summary: Extensions of Centrally Essential Rings

Problem Statement
The paper investigates the structural properties and extensions of centrally essential rings. A nonzero unital ring $R$ is defined as centrally essential if for every nonzero element $a \in R$ , there exist nonzero central elements $x, y \in Z(R)$ such that $ax = y$. This class of rings generalizes commutative rings; notably, it is established in prior literature (specifically [5, Proposition 2.8]) that semiprime centrally essential rings are necessarily commutative. Consequently, the authors focus on characterizing noncommutative centrally essential rings, specifically within the subclass of almost fully prime rings (rings where every nonzero proper ideal is prime). The central problem is to describe these noncommutative structures in terms of ideal extensions, Dorroh extensions, and trivial extensions.

Methodology
The authors employ a structural algebraic approach, utilizing the following definitions and tools:

Ideal Extensions: Analyzing rings $R$ containing an ideal $I'$ isomorphic to a ring $I$ such that $R/I' \cong Q$ .
Derivations: Utilizing the concept of incomparable derivations $d_1, d_2: F \to M$ (where kernels are not comparable) to construct noncommutative examples.
Ring Extensions:
- Dorroh Extension ( $D(R, S)$ ): Constructed from a ring $R$ and an $(R, R)$ -bimodule $S$ with multiplication $(r_1, s_1)(r_2, s_2) = (r_1r_2, r_1s_2 + s_1r_2 + s_1s_2)$ .
- Trivial Extension ( $T(R, M)$ ): A specific case where the bimodule multiplication is zero, yielding $(r_1, m_1)(r_2, m_2) = (r_1r_2, r_1m_2 + m_1r_2)$ .
Central Subsets: Defining specific subsets $T \subseteq R$ and $N \subseteq M$ based on the existence of nonzero central elements that map elements into the center or central submodules.

The proofs rely on analyzing the lattice of ideals, the properties of zero-divisors, and the interaction between the center of the ring and its ideals. The authors leverage results from previous works (e.g., [1], [2], [4], [5], [7]) regarding subdirectly indecomposable rings, uniform modules, and fully idempotent rings.

Key Contributions and Results

1. Characterization of Noncommutative Almost Fully Prime Centrally Essential Rings
The paper provides a comprehensive equivalence theorem (Theorem 1.2) for a noncommutative, almost fully prime ring $R$ . The following conditions are proven to be equivalent:

$R$ is a centrally essential ring.
$R$ possesses a unique nonzero proper ideal $P$ (which is contained in the center $Z(R)$ ), and the quotient ring $R/P$ is a field.
$R$ is an extension of a central ideal $I$ by a field $F$ .
$R$ is an extension of a central ideal $I$ by a field $F$ that admits two incomparable derivations from $F$ into some extension field.

This result implies that such rings are structurally very specific: they are subdirectly indecomposable, their set of zero-divisors coincides with the unique minimal ideal $P$ , and their additive group is isomorphic to a direct sum of copies of $\mathbb{Q}$ or $\mathbb{Z}_p$ (Proposition 4.2).

2. Conditions for Centrally Essential Extensions
The authors establish necessary and sufficient conditions for Dorroh and trivial extensions to preserve the centrally essential property (Theorem 1.3):

Dorroh Extensions: If $R$ is commutative, the Dorroh extension $D(R, S)$ is centrally essential if and only if the bimodule $S$ is centrally essential.
Trivial Extensions: The trivial extension $T(R, M)$ $T (R, M)$ is centrally essential if and only if two specific conditions hold:
1. $N = M$ (every element in the module can be mapped to a central element via a central ring element).
2. $T = R$ (every element in the ring can be mapped to a central element via a central ring element).

3. Embedding and Non-Unital Cases
The paper extends the definition to non-unital rings and proves (Corollary 3.1) that any centrally essential ring can be embedded into a centrally essential unital ring via the Dorroh extension over the integers, $D(\mathbb{Z}, R)$ .

4. Counter-Examples and Structural Nuances
The authors provide explicit examples to illustrate the boundaries of these definitions:

Example 1.1: Constructs a noncommutative almost fully prime centrally essential ring using a matrix ring over a field with incomparable derivations.
Example 4.5: Demonstrates a trivial extension $T(R, M)$ that is not centrally essential, showing that the conditions in Theorem 1.3 are strict.
Example 4.6: Constructs a centrally essential trivial extension where the left and right module actions differ ( $RM \neq MR$ ), highlighting that commutativity of the module action is not a prerequisite for the extension to be centrally essential.

Significance and Claims
The paper claims to provide a precise structural description of noncommutative almost fully prime centrally essential rings, linking them directly to field extensions with incomparable derivations. It clarifies the behavior of centrally essential properties under standard ring-theoretic constructions (Dorroh and trivial extensions). The work serves to refine the understanding of how centrality and essentiality interact in noncommutative settings, particularly distinguishing between rings that are "close" to commutative (via central ideals) and those that are not. The authors do not claim broad applications outside of ring theory but position the results as a foundational step in the classification of these specific algebraic structures.