Discrete Valuation Semirings

• Discrete Valuation Semirings are semiring analogues of discrete valuation rings, defined via a surjective valuation to ℤ and exhibiting a multiplicatively cancellative, principal ideal structure with a unique maximal ideal.
• They are characterized through equivalent properties such as being quasi-local, satisfying the ACCP, and having uniquely ordered ideals that enable a one-dimensional divisibility framework.
• Concrete constructions include localizations of factorial semirings and examples from nonnegative rational numbers under prime valuations, emphasizing their role in Gaussian and subtractive ideal theories.

Discrete valuation semirings (DVSs) are the semiring-theoretic counterparts of discrete valuation rings. In the formulation introduced by Nasehpour, a semiring $S$ is a discrete valuation semiring if $S$ is a $V$-semiring with respect to a triple $(K,v,\mathbb{Z})$, where $K$ is a semifield, $v$ is a surjective valuation, and $S=\{x\in K: v(x)\ge 0\}$. The theory places DVSs inside a broader valuation-semiring framework, proves that they are exactly the multiplicatively cancellative principal ideal semirings with a nonzero unique maximal ideal, and shows that subtractivity controls several ring-like properties, including integral closedness, the min-property for valuations, and Gaussianity (Nasehpour, 2015).

1. Valuation maps and semiring-theoretic preliminaries

Let $S$ be a commutative semiring with $0$ and $1$, and let $S$0 be a totally ordered commutative monoid with no greatest element. After adjoining a greatest element $S$1, one obtains $S$2 with $S$3. An $S$4-valuation on $S$5 is a map

$S$6

such that

$S$7

Associated subsets are

$S$8

When $S$9, the valuation is called discrete. The valuation is said to have the min-property if $V$0 implies

$V$1

Unlike the ring case, this is an additional condition rather than an automatic consequence (Nasehpour, 2015).

Two semiring-specific notions are central. An ideal $V$2 is subtractive if $V$3 and $V$4 imply $V$5. A semiring is multiplicatively cancellative (MC) if

$V$6

For an MC semiring $V$7, there is a semifield of fractions $V$8, and every element of $V$9 can be written as $(K,v,\mathbb{Z})$0 with $(K,v,\mathbb{Z})$1 and $(K,v,\mathbb{Z})$2. This fraction-semifield construction is the ambient setting in which valuation semirings and DVSs are formulated (Nasehpour, 2015).

2. Valuation semirings as the ambient class

A valuation semiring is defined by the existence of a surjective $(K,v,\mathbb{Z})$3-valuation $(K,v,\mathbb{Z})$4 on a semifield $(K,v,\mathbb{Z})$5 containing $(K,v,\mathbb{Z})$6 such that

$(K,v,\mathbb{Z})$7

In this situation, $(K,v,\mathbb{Z})$8 is an Abelian group, $(K,v,\mathbb{Z})$9 is MC, $K$0 is isomorphic to $K$1, and $K$2 is quasi-local with unique maximal ideal $K$3 (Nasehpour, 2015).

The structural theorem for valuation semirings is the semiring analogue of the classical characterization of valuation rings. For an MC semiring $K$4, the following are equivalent: $K$5 Thus the order-theoretic content of valuation theory survives intact in the MC-semiring setting: valuation semirings are precisely those semirings whose ideals are totally ordered by inclusion (Nasehpour, 2015).

A common misconception is that a weak valuation-like structure is already sufficient. It is not. Example 2.1 shows that $K$6, the Laurent polynomial semiring over the Boolean semiring $K$7, is a $K$8-semiring, but its principal ideals need not be totally ordered, since $K$9 and $v$0 are incomparable. This isolates the MC hypothesis and the total comparability of ideals as essential rather than cosmetic (Nasehpour, 2015).

3. Definition and equivalent characterizations of DVSs

A discrete valuation semiring is a $v$1-semiring with respect to $v$2, where $v$3 is a semifield and $v$4 is surjective. Equivalently, it is a valuation semiring whose value group is $v$5. The discreteness is therefore encoded in the fact that values occur in integral steps and that there exists an element $v$6 with $v$7 (Nasehpour, 2015).

The basic ideal structure is rigid. If $v$8 is a DVS, then there exists a nonzero and nonunit element $v$9 such that every nonzero ideal $S=\{x\in K: v(x)\ge 0\}$0 of $S=\{x\in K: v(x)\ge 0\}$1 has the form

$S=\{x\in K: v(x)\ge 0\}$2

This is obtained by choosing $S=\{x\in K: v(x)\ge 0\}$3 with $S=\{x\in K: v(x)\ge 0\}$4 and taking

$S=\{x\in K: v(x)\ge 0\}$5

for any nonzero proper ideal $S=\{x\in K: v(x)\ge 0\}$6 (Nasehpour, 2015).

The main characterization theorem gives six equivalent descriptions of a DVS. For a semiring $S=\{x\in K: v(x)\ge 0\}$7, the following are equivalent: $S=\{x\in K: v(x)\ge 0\}$8 This theorem makes precise that discreteness can be read equivalently from value-group data, ideal structure, factorization structure, or chain conditions on principal ideals (Nasehpour, 2015).

4. Subtractivity, integrality, Gaussianity, and Euclidean structure

Subtractive ideals are the additional layer that distinguishes semiring valuation theory from its ring-theoretic analogue. If $S=\{x\in K: v(x)\ge 0\}$9 is an MC semiring with fraction semifield $S$0, an element $S$1 is integral over $S$2 if there exist $S$3 such that

$S$4

The semiring is integrally closed if the elements of $S$5 integral over $S$6 are exactly the elements of $S$7. For valuation semirings, if the unique maximal ideal $S$8 is subtractive, then $S$9 is integrally closed (Nasehpour, 2015).

For DVSs, subtractivity is also equivalent to several other regularity properties. If $0$0 is a discrete valuation semiring, then the following are equivalent: $0$1 Here Gaussian means that for all $0$2, the content ideal satisfies

$0$3

Thus, in the discrete setting, “subtractive maximal ideal,” “min-property,” “all ideals subtractive,” and “Gaussian” are not separate phenomena but equivalent formulations of the same structural constraint (Nasehpour, 2015).

Two further consequences sharpen the picture. First, if $0$4 is an MC semiring satisfying ACCP and $0$5 is a nonunit, then

$0$6

a fact used in the DVS characterization. Second, any discrete valuation semiring is an Euclidean semiring. This places DVSs among the most rigid semirings in multiplicative ideal theory: all nonzero ideals lie in a single chain of powers, and the valuation furnishes an effective one-dimensional ordering of divisibility (Nasehpour, 2015).

5. Constructions and examples

One general construction begins with a factorial semiring $0$7, its semifield of fractions $0$8, and a fixed prime element $0$9. Every nonzero $1$0 can be written uniquely in the form

$1$1

with $1$2 and $1$3 not divisible by $1$4. Defining

$1$5

one obtains

$1$6

and $1$7 is a discrete valuation semiring. This is the semiring version of localization at a prime element in a factorial setting (Nasehpour, 2015).

A concrete instance is provided by the semifield $1$8 of nonnegative rational numbers. Fix a prime $1$9. Any nonzero $S$00 can be written uniquely as $S$01, where $S$02 and $S$03 are coprime to $S$04. Then

$S$05

defines a discrete valuation, and

$S$06

is a DVS. Moreover, $S$07 has the min-property, so $S$08 is a subtractive Gaussian DVS (Nasehpour, 2015).

A further source comes from Dedekind domains. If $S$09 is a Dedekind domain and $S$10 is a maximal ideal, then the ideal semiring $S$11, with operations $S$12 and $S$13, is a Gaussian factorial semiring. The subsemiring

$S$14

of the semifield of fractions $S$15 is a subtractive discrete valuation semiring. Remark 3.11 adds that Claborn’s theorem realizes arbitrary sets as sets of maximal ideals of Dedekind domains; this yields large families of Gaussian factorial semirings and corresponding DVSs (Nasehpour, 2015).

6. Position within valuation-type semiring theory

Subsequent work places DVSs inside a wider landscape of valuation-like semirings. A semiring is uniserial if its ideals are linearly ordered by inclusion, and a semidomain is a valuation semiring iff for every $S$16, either $S$17 or $S$18. The uniserial property is stable under matrix formation: $S$19 More generally, prime ideals of a commutative semiring are linearly ordered iff for each $S$20 there exists $S$21 such that either $S$22 or $S$23. In a semidomain, valuation structure can be recovered from the weaker spectral condition together with gcd theory: $S$24 Pseudo-valuation semidomains and divided semidomains enlarge this hierarchy; DVSs occupy the rigid end where full divisibility comparability and discreteness are both present (Behzadipour et al., 2018).

DVSs also fit naturally into the ideal-theoretic theory of Prüfer and Gaussian semirings. A Prüfer semiring is one in which every nonzero finitely generated ideal is invertible. For a semidomain $S$25,

$S$26

and equivalently for every maximal ideal $S$27. In a local semidomain,

$S$28

Under the additional hypotheses that $S$29 is subtractive and $S$30 for all $S$31, Prüfer and Gaussian are equivalent. This situates DVSs as local valuation objects inside a broader Prüfer-Gaussian ideal theory (Ghalandarzadeh et al., 2014).

A parallel valuation-theoretic program on idempotent and tropical semirings introduces classical valuations, strict valuations, and hyperfield valuations. For $S$32, nontrivial strict valuations trivial on $S$33 yield value groups isomorphic to $S$34, and nontrivial hyperfield valuations trivial on $S$35 are classified by

$S$36

The associated local semirings at points of the resulting abstract curves are valuation semirings with value group $S$37. This suggests that tropical and hyperfield geometry provide a natural geometric habitat for DVS-like local objects (Jun, 2015).

On the ring side, the formalization of complete discrete valuation rings and local fields in Lean proves that the unit ball with respect to a discrete valuation on a field is a discrete valuation ring, and conversely that the adic valuation on the field of fractions of a discrete valuation ring is discrete. A plausible implication is that an analogous formalization of DVSs would proceed by emphasizing unit-ball constructions, valuation subsemirings, and the discrete value group $S$38 (Frutos-Fernández et al., 2023).