Finite Local PIR: Structure & Classification

• Finite local PIRs are finite commutative rings with a unique maximal ideal and all ideals principal, characterized by a nilpotency index and a ramification index.
• The classification relies on key invariants (p, q, e, n, f) and can be explicitly constructed via quotients of discrete valuation rings using Eisenstein polynomial orbits.
• Graph-theoretic and module-theoretic approaches provide practical insights, offering categorical invariants and decomposing the ring structure into staircase graphs and additive components.

A finite commutative local principal ring, abbreviated here as "finite local PIR," is a finite commutative ring with unity that is both local (possessing a unique maximal ideal) and a principal ideal ring (every ideal is generated by a single element). Such rings are fundamental in commutative and algebraic ring theory, serving as a key class of Artinian rings, and presenting a structure that is explicitly computable yet richly connected to local field theory, valuation theory, and module classification.

1. Structural Definition and Fundamental Properties

Let $R$ be a finite commutative ring with unity. $R$ is local if it contains a unique maximal ideal $\mathfrak{m}$. $R$ is a principal ideal ring if every ideal is principal. In a finite local PIR, the ideals form a strictly descending chain: $$0 = \mathfrak{m}^n \subset \mathfrak{m}^{n-1} \subset \cdots \subset \mathfrak{m} \subset R$$ with each $\mathfrak{m}^i$ principal. Two principal invariants arise from this chain: the nilpotency index $n$, defined by $\mathfrak{m}^n = 0$ and $\mathfrak{m}^{n-1} \ne 0$, and the ramification index $e = v_R(p)$, corresponding to the valuation of the characteristic prime $p \in \mathfrak{m}$.

Letting $k = R/\mathfrak{m}$ (the residue field), the following hold:

• The characteristic of $k$ is a prime $p$, and $|k| = q = p^f$.
• For any uniformizer $\pi \in R$, $p = u \pi^e$ for some unit $u \in R^\times$.
• The characteristic of $R$ is $p^a$, with $a = \lceil n/e \rceil$.

The five invariants determining $R$ up to a finer classification are: $$p,\quad q = |k|,\quad e,\quad n,\quad f = [k : \mathbb{F}_p]$$ A single additional invariant—the orbit of a suitable Eisenstein polynomial under a group action—is required for complete isomorphism classification (Lee, 2023, Wu et al., 2011).

2. Classification and Construction

Every finite local PIR arises as a quotient of a discrete valuation ring (DVR) associated to a totally ramified extension over a number field. Explicitly, given invariants $(p, q, e, n, f)$, one can construct:

• A number field $K$ of degree $f$ over $\mathbb{Q}$ in which $p$ is inert.
• A totally ramified extension $L / K$ of degree $e$, with rings of integers $\mathcal{O}_K \subset \mathcal{O}_L$.
• The complete DVR $\mathcal{D} = (\mathcal{O}_L)_\mathcal{P}$ at the unique prime $\mathcal{P}$ above $p$.

The local PIR is then

$R \cong \mathcal{D}/(\pi^n)$

where $\pi$ is a uniformizer of $\mathcal{D}$. The isomorphism type of $R$ is determined by the five invariants and a Galois-cohomological datum: the $G$-orbit of an Eisenstein polynomial of degree $e$ over the unramified coefficient ring (Lee, 2023).

In equal characteristic ($\text{char}(R) = p$), every such $R$ is isomorphic to a truncated polynomial ring: $R \cong k[x]/(x^e)$ with $x$ a generator of the maximal ideal. In the mixed-characteristic case ($\text{char}(R) = p^r$, $r \geq 2$), the structure is of the form

$R \cong S[x]/(x^e,\, p - u x^t)$

where $S$ is the unramified extension $W_r(\mathbb{F}_{p^f})$ and $u \in S^\times$, $1 \leq t \leq e$ (Wu et al., 2011).

3. Invariant Data and Eisenstein Orbits

Besides the elementary ring invariants, the complete classification up to isomorphism is given by the action of the group $T = (W[Y]/(Y^n))^\times$ on Eisenstein polynomials of degree $e$ over the Cohen/Witt ring $W(\mathbb{F}_q)$. The essential set is

$\Eis_e(W) = \big\{ f(Y) = Y^e + a_{e-1}Y^{e-1} + \cdots + a_0 \;\mid\; a_i \in (\varpi),\; a_0 \not\in (\varpi)^2 \big\}$

with the group action defined by $t \star f = \operatorname{Res}_Y(f(Y), Z - t(Y)Y)$, interpreted modulo $Y^n$ and resulting in a new Eisenstein polynomial. Isomorphism classes of finite local PIRs of a given type $(p, q, e, n)$ correspond bijectively to the orbits in $\Eis_e(W)/T$ (Lee, 2023).

4. Graph-Theoretic Characterization

Compressed zero-divisor graphs provide a categorical invariant distinguishing finite local PIRs. Define the compressed zero-divisor graph $\Theta(K)$ for a finite commutative unital ring $K$:

• Vertices are associatedness classes $[a]$, where $a \sim b$ if $a = bu$ for a unit $u$.
• Edges connect $[a]$, $[b]$ if $ab = 0$.

A ring $K$ is a local PIR if and only if $\Theta(K)$ is isomorphic to a staircase graph $SG_n$ (for some $n$), characterized by a unique vertex for each degree $1,2,\ldots,n+1$ and a strictly increasing degree sequence. The nilpotency index of the maximal ideal equals the length $n$ of the staircase (Đurić et al., 2018). The graph-theoretic approach is functorial and detects locality as indecomposability of the graph.

Class of Ring	Graph Structure	Key Invariant
Local PIR ($K$)	$SG_n$	Nilpotency index $n$
General PIR	Product of $SG_{k_i}$	Factors $(k_i)$

5. Numerical and Module-Theoretic Invariants

For a finite local PIR $R$ with residue field $k \cong \mathbb{F}_{p^f}$, maximal ideal of nilpotency index $e$, and characteristic $p^r$:

• $|R| = p^{\,r + f(e-1)}$ (if $r=1$); more generally, $|R| = |S|^e = (p^r)^{fe}$ for $S = W_r(\mathbb{F}_{p^f})$.
• The additive group of $R$ decomposes as $S \oplus Sx \oplus \ldots \oplus Sx^{e-1}$, for $x$ a uniformizer.
• Nonzero ideals are exactly $(x), (x^2), \ldots, (x^{e-1})$.
• The characteristic is constrained by $1 \leq r \leq e$ (Wu et al., 2011).

In all cases, the remaining “shape” of the ring—distinguishing non-isomorphic rings with identical invariants—traces to the orbit structure of Eisenstein polynomials or, graph-theoretically, to the unique structure of the staircase graph.

6. Illustrative Examples

Equal characteristic ($r=1$):

$R \cong \mathbb{F}_{p^f}[x]/(x^e)$

is uniquely determined by $(p, f, e)$.

Mixed characteristic ($r \geq 2$):

$$R \cong S[x]/(x^e,\, p-u x^t),\quad S = W_r(\mathbb{F}_{p^f})$$

Admissible parameter choices for $u$ and $t$ distinguish isomorphism types.

Graph characterization:

For $R = \mathbb{Z}/16$, $\Theta(R) \cong SG_3$ and the ring is a local PIR with nilpotency index $4$. For more general rings, the shape and multiplicity of staircases in $\Theta$ reflect the decomposition and principal structure (Đurić et al., 2018).

Eisenstein polynomial orbit:

For quadratic $e=2$, $n=2$, over $W = \mathbb{Z}_p$, two orbits appear, yielding two non-isomorphic local PIRs of type $(p, p, 2, 2)$, distinguished by the shape of the Eisenstein relation (Lee, 2023).

7. Broader Context and Generalizations

Finite commutative local PIRs are precisely the finite local Artinian rings with principal maximal ideal (Wu et al., 2011). They provide local building blocks in the primary decomposition of finite commutative PIRs and appear as homomorphic images of DVRs truncated at a power of a uniformizer. Their explicit construction bridges the structure theory of commutative rings, valuation theory, and Galois theory via the subtle classification using Eisenstein polynomial orbits. Extensions to infinite residue field or artinian cases, and links to tamely and wildly ramified extensions, naturalize in the language of complete local rings and Cohen structure theory—a plausible implication is that much of the combinatorial and structural apparatus developed for the finite case extends, with modifications, to a broader (possibly infinite) context (Lee, 2023, Wu et al., 2011).