---
title: Rings
description: Rings are algebraic structures equipped with two binary operations generalizing addition and multiplication. Learners will understand ring axioms, ideals, and homomorphisms, enabling them to analyze polynomial rings, fields, and factorization domains.
category: mathematics
subcategory: abstract-algebra
difficulty: beginner, intermediate, advanced
url: /subject/rings
---

# Rings

Rings are algebraic structures equipped with two binary operations generalizing addition and multiplication. Learners will understand ring axioms, ideals, and homomorphisms, enabling them to analyze polynomial rings, fields, and factorization domains.

## Available Resources

2 Books • 1 Courses • 2 Websites

## Websites

### 1. nLab: Ring

The nLab reference entry on rings, defining them as monoids in the category of abelian groups and relating them to rigs, algebras, and modules. Useful for readers who know basic ring theory and want its categorical formulation and generalizations.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/ring

**Tags:** ring-theory, category-theory, abstract-algebra, monoids

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Abstract Algebra

Recorded lectures from Benedict Gross's Harvard abstract algebra course, following Artin's Algebra. Covers groups, symmetry, linear groups, and bilinear forms before moving to rings, ideals, factorization, and fields, preparing viewers for graduate-level algebra.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5

**Tags:** abstract-algebra, ring-theory, group-theory, field-theory, lectures

## Books

### 1. A Book of Abstract Algebra - Charles C. Pinter

**Author:** Charles C. Pinter

Pinter's Dover introduction to abstract algebra, with short chapters that develop groups, rings, integral domains, and fields through many graded exercises. Readers learn to write proofs about algebraic structures and reach Galois theory without prior abstract mathematics.

**Difficulty:** Beginner | **Language:** English | **Price:** Paid

**Link:** https://amzn.to/465eXss

**Tags:** abstract-algebra, group-theory, ring-theory, field-theory

### 2. Introduction to Abstract Algebra

**Author:** W. Keith Nicholson

An undergraduate textbook covering groups, rings, polynomials, and fields, with chapters on factorization in integral domains, field extensions, and Galois theory. Readers learn to work with ideals, quotient rings, and homomorphisms and prove standard structure theorems.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0471694924?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, abstract-algebra

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