Left Ideal Covering Number in Rings

• Left ideal covering number is defined as the minimal cardinality of proper left ideals whose union equals a ring, generalizing classical group covering concepts.
• Key results classify ηℓ-elementary rings, notably those isomorphic to block matrices over finite fields with covering number (q^(n+1) - 1)/(q - 1).
• Applications include structural insights into noncommutative rings, implications for module theory, and connections to combinatorial and linear algebra covering problems.

The left ideal covering number, denoted $\eta_\ell(R)$, is a cardinal invariant of an associative ring $R$—not assumed to be commutative or unital—which measures the minimal cardinality of a set of proper left ideals whose union equals $R$. This concept generalizes the classical covering number of groups and connects with covering numbers in linear algebra, providing insight into the interplay between additive and multiplicative structure in rings.

1. Formal Definition and Basic Properties

Given an associative ring $R$, a cover by left ideals is a finite collection $\{L_1, L_2, \ldots, L_m\}$ of proper left ideals such that

$R = \bigcup_{i=1}^m L_i$

If such a cover exists, the minimal possible cardinality $m$ is called the left ideal covering number $\eta_\ell(R)$. If no such cover exists—most notably when $R$ has a multiplicative identity—the invariant is set to $\infty$ since any ideal containing 1 must be the entire ring.

In parallel, the minimal covers by right and two-sided ideals are denoted $\eta_r(R)$ and $\eta(R)$, respectively.

The covering number of the additive group $R^+$, denoted $\sigma(R^+)$, often serves as a lower bound for $\eta_\ell(R)$, with the following chain of inequalities holding for suitable $R$:

$$\sigma(R^+) \leq \sigma(R) \leq \eta_\ell(R), \eta_r(R) \leq \eta(R)$$

2. Structural Classification and Key Results

The classification of rings with finite left ideal covering number is governed by the concept of $\eta_\ell$-elementary rings. A ring $R$ is said to be $\eta_\ell$-elementary if for every nonzero two-sided ideal $I$ of $R$, one has $\eta_\ell(R) < \eta_\ell(R/I)$; that is, $R$ achieves the minimal possible covering number among its quotients.

Main Classification Theorem

A noncommutative ring $R$ is $\eta_\ell$-elementary if and only if

$$R \cong \{(A \mid v): A \in M_n(F_q),\ v \in F_q^n\} \subset M_{n+1}(F_q)$$

where $F_q$ is a finite field of order $q$ and $n \geq 1$. The elements are block matrices of the form

$$\begin{bmatrix} A & v \ 0 & 0 \end{bmatrix}$$

with $A \in M_n(F_q)$, $v \in F_q^n$.

The left ideal covering number of such rings is

$\eta_\ell(R) = \frac{q^{n+1} - 1}{q - 1}$

corresponding to the number of $1$-dimensional subspaces of $F_q^{n+1}$.

Any finite ring admitting a finite left ideal cover can be “reduced” via a suitable two-sided ideal to a residue ring which is $\eta_\ell$-elementary, thus the classification of $\eta_\ell$-elementary rings identifies all rings with finite left ideal covering number (Chen et al., 23 Sep 2025).

3. Explicit Constructions and Extreme Cases

Central examples of rings with small ideal covering numbers are constructed for each prime $p$; these realize the minimal possible covering numbers and exhibit both symmetry and asymmetry in their left/right/two-sided properties (Chen, 7 Aug 2025):

Family	$\eta_\ell$	$\eta_r$	$\eta$	Presentation (for prime $p$)
$R_1$	$p + 1$	$p + 1$	$p + 1$	$a^2 = b^2 = ab = ba = 0$
$R_2$	$p + 1$	$\infty$	$\infty$	$a^2=a,\ b^2=b,\ ab=b,\ ba=a$
$R_3$	$\infty$	$p + 1$	$\infty$	$a^2=a,\ b^2=b,\ ab=a,\ ba=b$
$R_4$	$p + 1$	$p + 1$	$\infty$	See section 3 for relations

These families demonstrate that the additive structure (often $C_p \times C_p$) does not guarantee covering by two-sided ideals, and that the multiplication rules can force asymmetry between left and right ideal covers.

For rings of order $p^2$ with additive group $C_p \times C_p$, all $p+1$ subgroups of order $p$ are candidates for ideal covers, but whether these are ideals depends on the multiplicative structure.

The classification for covering number $3$ is explicit: a ring $R$ has $\eta_\ell(R) = 3$ precisely when it has a factor ring isomorphic to a specific $4$-element subring of $M_2(\mathbb{Z}_2)$ or $M_2(\mathbb{Z}_4)$ as formalized in (Chen, 7 Aug 2025).

4. Methodologies and Connections to Linear Algebra

The results draw upon methods from both ring theory and linear algebra. In particular, the techniques of lifting covers via quotient maps, common in vector space covering problems (Clark, 2012), translate to rings in finding covers of larger structures from smaller “elementary” ones.

The connection is explicit: for a vector space $V$ of dimension $2$ over field $K$, the linear covering number is $|K|+1$; for a ring $R$ whose additive group is analogous to $V$, similar bounds are observed for $\eta_\ell(R)$, with the minimal irredundant covering also rigid at $|K|+1$ in this setting.

Additionally, the decomposition $R = S \oplus J$ (with $S$ simple semisimple, $J = J(R)$ the Jacobson radical) leverages classical ring-theoretic arguments, including embeddings into Dorroh extensions and Nakayama's Lemma for controlling ideal structure (Chen et al., 23 Sep 2025).

5. Applications and Implications

The paper of covering rings by left ideals generalizes classical group covering problems to nonunital and noncommutative rings, exposing a rich spectrum of behaviors depending on the interplay of additive and multiplicative structures.

• Structural insight: The existence of a finite left ideal cover imposes strong restrictions: $R$ must be finite of prime characteristic, often with a strictly constrained decomposition. For noncommutative cases, rings with $\eta_\ell(R) < \infty$ are constructed explicitly, and module-theoretic analogues obtain covering numbers as in the vector space case.
• Module theory: The covering number of modules over matrix rings $M_n(F_q)$ is determined explicitly, and every such module can be covered by $\frac{q^{n+1} - 1}{q - 1}$ proper submodules; this generalizes previous work for modules over commutative rings.
• Combinatorial algebra: Knowledge of ideal covering numbers influences understanding of ring decompositions and has implications on related questions in combinatorial algebra, such as subgroup covers and normal covers in group theory.

6. Open Questions and Future Directions

Several questions remain unresolved:

• Determination of covering numbers for modules over noncommutative rings: the only known values are those given by the matrix ring construction, leaving open the existence of other values.
• Precise conditions for coverability by proper subrings and the possible forms of ideal covering numbers: are all such numbers of the form $p^d + 1$, and if so, what dictates $d$ for two-sided covers?
• Whether every ring with minimal covering number $p + 1$ has a factor ring isomorphic to one of the model rings identified above.
• Extension of these results to infinite rings, rings with additional structure, or computational approaches, which are suggested by current investigations using software such as GAP.

7. Comparative Context and Broader Significance

The concept of the left ideal covering number situates itself at an intersection of classical covering problems for groups and vector spaces, and modern ring theory. The results attained provide comprehensive classification and explicit realization of minimal coverings in finite and nonunital rings, clarifying the combinatorial invariants that control such decompositions.

The work referenced here achieves a complete characterization of rings admitting finite left ideal covers, demonstrates sharp lower bounds for covering numbers in various contexts, and establishes connections with parallel invariants in module and linear algebra theory (Chen et al., 23 Sep 2025, Chen, 7 Aug 2025, Clark, 2012). Open questions both in module theory and combinatorial algebra ensure the topic remains a fertile ground for further research.