Engineering / Mathematics — Rings and Polynomial Rings
Rings: Definitions, Properties and Examples
Commutative rings with unity: axioms, units, and the standard examples used throughout the subject.
Executive summary
A ring has two operations, addition making it an abelian group and multiplication distributing over it. Restricting to commutative rings with a multiplicative identity covers everything needed here.
The units of a ring form a group, which links the ring theory back to the group theory and explains why the two streams interlock.
Learning objectives
- State the ring axioms in the commutative-with-unity setting.
- Identify the units of a ring.
- Recognise the standard ring examples.
01The axioms
Commutative ring with unity
A set R with operations + and · such that:
(R, +) is an abelian group with identity 0;
· is associative and commutative with identity 1;
distributivity holds: a(b + c) = ab + ac.
Two immediate consequences: 0 · a = 0 for every a, proved from distributivity, and (−a)b = −(ab). Neither is an axiom.
02Units
Unit
a ∈ R is a unit if there is b ∈ R with ab = 1. The units form a group under multiplication, written R*.
| Ring | Units | Structure |
|---|---|---|
| Z | {1, −1} | Cyclic of order 2 |
| Z_n | {a : gcd(a,n) = 1} | Order φ(n) |
| A field F | F minus {0} | Order |F| − 1 |
| F[X] | Non-zero constants | F* |
| F[[X]] | Series with non-zero constant term | Infinite |
The fourth row is worth noting: in a polynomial ring over a field, only the non-zero constants are invertible. No polynomial of positive degree has an inverse, because degrees add under multiplication and cannot cancel.
03The standard examples
The integers Z
The prototype. An integral domain, not a field, with unique factorisation.
Integers modulo n
Finite; a field exactly when n is prime, otherwise carrying zero divisors.
Polynomial rings F[X]
Behaves remarkably like Z: division with remainder, Euclid, unique factorisation.
Finite fields F_q
The setting for coding theory and much of the algorithmic content.
The parallel between Z and F[X] is not superficial. Both are Euclidean domains, and essentially every algorithm developed for integers — Euclid, extended Euclid, Chinese remaindering, rational reconstruction — transfers to polynomials with degree replacing absolute value.
04Frequently asked questions
Must a ring have a multiplicative identity?
Not under the general definition, but it is assumed throughout this collection. Rings without identity are called rngs by some authors and are not needed here.
Is every field a ring?
Yes — a field is a commutative ring with unity in which every non-zero element is a unit. The hierarchy runs ring, integral domain, field, each strictly narrower.
Why exclude non-commutative rings?
Because nothing in this subject requires them and the theory is substantially more complex. Every ring arising here — integers, modular integers, polynomials, finite fields — is commutative.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 211-215.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.