Ordered semigroups and ideal categories of principal ideal rings

Technical Summary: Ordered Semigroups and Ideal Categories of Principal Ideal Rings

Problem Statement
The paper investigates the algebraic and categorical structures inherent in the collection of ideals of a commutative principal ideal ring (PIR) with unity. Specifically, the authors address the dual nature of this collection: first, as an ordered semigroup $I(R)$ under ideal multiplication and a divisibility-based partial order, and second, as a category $I_R$ where objects are ideals and morphisms are $R$-linear maps. The central problem is to characterize the properties of the ordered semigroup $I(R)$ and to establish a rigorous correspondence between this semigroup structure and the categorical structure of $I_R$.

Methodology
The authors employ a combination of ordered semigroup theory, ring theory, and category theory.

1. Ordered Semigroup Construction: The set of ideals $I(R)$ is equipped with ideal multiplication as the semigroup operation. A partial order $\preccurlyeq$ is defined on $I(R)$ such that for principal ideals $A=\langle a \rangle$ and $B=\langle b \rangle$, $A \preccurlyeq B$ if and only if $a$ divides $b$ in $R$. The authors verify that this order is compatible with multiplication and that every element in $I(R)$ is "ordered regular" (satisfying $A \preccurlyeq AXA$ for some $X$).
2. Categorical Formulation: The category $I_R$ is defined with ideals as objects and $R$-linear maps as morphisms. The authors identify a subcategory $P$ consisting of inclusion maps, demonstrating that $(I_R, P)$ satisfies the axioms of a "category with subobjects."
3. Structural Correspondence: To link these structures, the authors construct a strict preorder category $C$ where a morphism exists from $A$ to $B$ if and only if $A \preccurlyeq B$. They then define a contravariant functor $F: C \to I_R$ that maps the order relation in the semigroup to inclusion maps in the category.

Key Results and Contributions
The paper establishes several specific properties and theorems regarding the structure of ideals in commutative PIRs:

• Regular Ordered Semigroup Structure: Theorem 2.1 proves that $(I(R), \cdot, \preccurlyeq)$ is a regular ordered semigroup. Unlike classical regularity which requires $a = axa$, ordered regularity here requires $a \preccurlyeq axa$. The authors show that for any ideal $A=\langle a \rangle$, $A \preccurlyeq A \langle x \rangle A$ holds for any $x \in R$.
• Strong Regularity Properties: Theorem 2.10 demonstrates that $I(R)$ satisfies a robust set of regularity conditions:
  • It is intra-regular ($A \preccurlyeq XA^2Y$).
  • It is an inverse ordered semigroup (every element has a unique ordered inverse up to $\mathcal{H}$-relation).
  • Every element is an ordered idempotent ($A \preccurlyeq A^2$).
  • It is completely regular, group-like, and a Clifford ordered semigroup.
• Green's Relations: Theorem 2.9 establishes that all Green's relations ($\mathcal{L}, \mathcal{R}, \mathcal{J}, \mathcal{H}, \mathcal{D}$) on $I(R)$ are universal relations. This is a consequence of the existence of the zero ideal and the commutativity of $R$, which forces every ideal to be related to every other ideal under these equivalence relations.
• Isomorphism and Inverse Transversals:
  • Theorem 2.5 shows that if two commutative PIRs $R$ and $S$ are isomorphic, their ideal semigroups are isomorphic. The converse is noted to not hold generally (e.g., fields $\mathbb{R}$ and $\mathbb{Q}$).
  • Proposition 2.7 identifies the set containing only the unit ideal $\{\langle 1 \rangle\}$ as an inverse transversal for the ordered semigroup of ideals.
  • Proposition 2.8 links the von Neumann regularity of an element $a \in R$ to the regularity of the principal ideal $\langle a \rangle$ in the semigroup $I(R)$.
• Concrete Examples: The paper provides explicit isomorphisms for specific rings:
  • $I(\mathbb{Z})$ is isomorphic to the ordered semigroup of non-negative integers under multiplication and divisibility.
  • $I(\mathbb{Z}_n)$ is isomorphic to the set of divisors of $n$ with a specific gcd-based operation.
  • $I(F[x])$ is isomorphic to the set of equivalence classes of polynomials under scalar multiplication, ordered by divisibility.
• Categorical Correspondence: The primary theoretical contribution is the construction of a contravariant functor $F$ from the preorder category $C$ (derived from the divisibility order of $I(R)$) to the category $I_R$ (ideals with linear maps). This functor maps the existence of an order relation $A \preccurlyeq B$ to the inclusion morphism $j(B, A): B \to A$ in the category of ideals.

Significance and Claims
The paper claims to provide a unified perspective on the ideals of a commutative principal ideal ring by viewing them simultaneously as an ordered semigroup and a category with subobjects. The significance lies in the demonstration that the divisibility of generators induces a structure where the semigroup is not merely regular but satisfies strong inverse and Clifford properties. Furthermore, the authors establish a natural correspondence between the ordered semigroup structure and the categorical structure via a contravariant functor, effectively translating the order-theoretic properties of ideals into categorical inclusion maps. The work contributes to the broader investigation of ordered semigroups and the categorical representation of algebraic structures, specifically clarifying the behavior of ideals in PIRs within these frameworks.