Rings vs semirings

Rings vs semirings

Overview A semiring is an algebraic structure with two binary operations (usually called “addition” and “multiplication”) satisfying most of the familiar ring axioms except that additive inverses need not exist. A ring, by contrast, does require that every element have an additive inverse. Below is a side-by-side comparison of their defining properties, followed by examples and some remarks on when each arises in mathematics and computer science.

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1. Definitions and Axioms

Property	Semiring	Ring
Underlying set	$S$	$R$
Addition $(+,0)$	Commutative monoid:$\forall a,b,c\in S$,$a+(b+c)=(a+b)+c$,$a+b=b+a$,$0+a=a$.	Abelian group:All semiring properties plus$\forall a\exists(-a)$ such that $a+(-a)=0$.
Multiplication $(\cdot,1)$	Monoid:$\forall a,b,c\in S$,$a\cdot(b\cdot c)=(a\cdot b)\cdot c$,often with identity $1$.	Monoid:Same as semiring. Often required to have $1$.
Distributivity	$\forall a,b,c$,$a\cdot(b+c)=a\cdot b + a\cdot c$,$(b+c)\cdot a = b\cdot a + c\cdot a$.	Same as semiring.
Zero-absorption	$\forall a,;0\cdot a = a\cdot 0 = 0$.	Same as semiring.
Additive inverses	Not required	Required

Note on terminology:

• Some authors require a multiplicative identity $1$ in both structures; others call a structure without $1$ a “hemiring” (for semirings without identity) or a “rng” (for rings without unity).
• In this summary we assume both have $1$, unless noted otherwise.

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2. Key Differences

1. Additive inverses

   • Ring: Every element $a$ has an inverse $-a$ so that $a + (-a) = 0$.
   • Semiring: May lack inverses; subtraction need not be defined.

2. Examples

   • Semiring but not Ring:

     • $\mathbb{N} = {0,1,2,\dots}$ with usual $+$ and $\times$.
     • The Boolean semiring ${0,1}$ with $+$ = OR, $\cdot$ = AND.

   • Ring:

     • $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$.
     • Matrix rings $M_n(\mathbb{R})$.

3. Subtraction vs. cancellation

   • In rings you can “cancel” addition using inverses: from $a + x = a + y$ infer $x=y$.
   • In semirings, one often works with cancellative semirings to get a weak form of cancellation, but it’s not automatic.

4. Applications

   • Semirings:

     • Automata theory & formal languages (e.g.\ shortest-path semirings).
     • Tropical geometry (min-plus semiring).
     • Quantitative program analysis (probabilities, costs).

   • Rings:

     • Core algebra and number theory.
     • Algebraic geometry, homological algebra.

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3. Examples in More Detail

1. Natural numbers $(\mathbb{N}, +, \times)$

   • Satisfies all semiring axioms: $0$ is additive identity, $1$ is multiplicative identity.
   • Fails ring axioms because there is no $n\in\mathbb{N}$ with $3 + n = 0$.

2. Tropical semiring

   • Underlying set $\mathbb{R}\cup{\infty}$.
   • “Addition” is $\min$, “multiplication” is $+$.
   • Used for shortest-path computations (Dijkstra, Bellman–Ford).

3. Boolean semiring ${0,1}$

   • $a+b = \max(a,b)$ (logical OR), $a\cdot b = \min(a,b)$ (logical AND).
   • Fundamental in logic, circuit design, and automata.

4. Integer ring $(\mathbb{Z}, +, \times)$

   • Every integer has an additive inverse.
   • Ring of central importance in algebra and number theory.

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4. When to Use Which?

• Use a semiring when only nonnegative (or otherwise “one-sided”) quantities make sense and subtraction would be inappropriate or undefined.
• Use a ring when a full-fledged additive group structure (including negatives) is needed for constructions like modules, ideals, and homological tools.

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Further Reading

• Golan, J. S. Semirings and their Applications (Kluwer, 1999)
• Dummit & Foote, Abstract Algebra (especially the sections on rings and modules)

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By focusing on the presence or absence of additive inverses, you can quickly tell whether an algebraic system is a semiring or a ring—and choose the right framework for your application.