Ring (mathematics)

In mathematics, a ring is a special kind of algebraic structure. It has two main operations: addition and multiplication. These operations work like the addition and multiplication you know from regular numbers, but with one difference: in a ring, multiplying numbers doesn’t always give the same result, depending on the order.

Ring elements can be numbers like integers or complex numbers. They can also be other things, like polynomials, square matrices, functions, and power series.

A ring is defined more simply as a set where addition works like an abelian group, multiplication is associative and distributive over addition, and there is a special element called the multiplicative identity element. When multiplication in a ring gives the same result no matter the order, it is called a commutative ring. Commutative rings are important because they help us understand many areas of mathematics, like algebraic number theory and algebraic geometry.

Examples of commutative rings include the real numbers, the integers, and polynomials. Examples of noncommutative rings include the ring of n × n real square matrices where n is 2 or more. The idea of rings was developed a long time ago by mathematicians such as Richard Dedekind, David Hilbert, Abraham Fraenkel, and Emmy Noether. Today, rings are a key part of many areas of mathematics.

Definition

A ring is a special group of numbers or objects. It has two ways to combine them, called addition and multiplication. These ways of combining work like the addition and multiplication you know, but they follow some important rules.

For addition in a ring, it must work like normal addition. It must be associative, meaning the order you add doesn't change the result. It must also be commutative, meaning the order of the numbers doesn’t change the result. There must also be an additive identity, a number you can add to any other number without changing its value.

Multiplication in a ring must also be associative, and there must be a multiplicative identity. Multiplication must also distribute over addition. This means multiplying a number by a sum is the same as multiplying the number by each part of the sum and then adding those results together.

set binary operations abelian group additive identity additive inverse monoid multiplicative identity distributive rng even integers commutative rings nonzero multiplicative inverse division ring field nonassociative ring algebra

Illustration

The most familiar example of a ring is the set of all integers. This includes numbers like ..., -5, -4, -3, ..., 0, 1, 2, 3, .... These numbers follow special rules when we add or multiply them, just like how we usually work with numbers.

One interesting example is "integers modulo 4." Here, we only use the remainders after dividing by 4: 0, 1, 2, and 3. We can still add and multiply these remainders, but we always take the remainder after dividing by 4 again. For example, adding 2 and 3 gives us 1 (since 2 + 3 = 5, and 5 divided by 4 leaves a remainder of 1). Multiplying 2 and 3 also gives us 2 (since 2 × 3 = 6, and 6 divided by 4 leaves a remainder of 2).

Another example involves 2-by-2 square matrices. These are grids with four numbers, like: $$ \begin{pmatrix} a & b \ c & d \end{pmatrix} $$ We can add and multiply these matrices in special ways, and they still follow the ring rules. For instance, the matrix $$ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $$ acts like the number 1 because multiplying it by any other matrix leaves that matrix unchanged.

History

See also: Ring theory § History

The idea of rings started from studying patterns in numbers and equations. In 1871, Richard Dedekind looked at special number patterns and created ideas that helped shape what we now call rings.

Later, in 1892, David Hilbert used the term “number ring” to describe certain number patterns. Around 1915, Abraham Fraenkel created the first strict rules for what a ring should be. In 1921, Emmy Noether improved these rules and helped us understand rings better today.

Over time, mathematicians have talked about whether rings should always include a special number called “1”. Some books say yes, others say no.

Basic examples

See also: Associative algebra § Examples

Commutative rings

The simplest example of a ring is the set of integers. We use the usual addition and multiplication. Other examples are the rational numbers, real numbers, and complex numbers. These are special types of rings called fields. We can also create rings using polynomials, like (x^2 + 2x + 1). The numbers in these expressions come from another ring. For example, polynomials with integer numbers form a ring. Functions that use real numbers and give real numbers can also form a ring under some rules.

Noncommutative rings

Not all rings work the same way as with normal numbers. For example, square matrices — grids of numbers — can form a ring. Their multiplication uses a special rule. If the matrices are bigger than one by one, their multiplication order is different from regular numbers. This makes these rings noncommutative. Another example is the set of ways to change or map elements of some math structures. These can also form rings with their own rules for addition and multiplication.

Non-rings

Some sets with addition and multiplication, like the natural numbers (1, 2, 3, and so on), are not rings. They lack some properties, such as numbers that can “undo” addition (like negative numbers for integers). We can expand natural numbers to include negatives, forming the ring of integers. There are also special cases where multiplication works in a non-standard way. These lead to structures similar to rings but not quite rings themselves.

Basic concepts

A ring is a special kind of mathematical structure. It has two operations, like adding and multiplying numbers. These operations work a bit like they do with regular numbers, but there is a twist: multiplication might not commute. This means that in some rings, switching the order of multiplication can change the result.

Rings can include many different kinds of objects, not just numbers. For example, you can have rings made of integers, polynomials, or even matrices. Each of these has its own interesting properties and uses in mathematics.

Modules

Main article: Module (mathematics)

Modules are like vector spaces, but they use elements from a ring instead of numbers for multiplication. A module is a special group that follows certain rules when its elements are multiplied by elements of the ring. These rules show how the elements work together.

One big difference between modules and vector spaces is that modules don’t always have a “size.” This makes studying modules more complex. For example, not all modules have a basis — a set of elements that can build every other element in the module. Modules help us understand many areas of mathematics by showing how different structures can be built from rings.

Constructions

Main article: Direct product of rings

A ring is a special math structure with two operations: addition and multiplication. These operations are like adding and multiplying whole numbers, but they can work a little differently. For example, in a ring, multiplication might not change if you switch the order.

Rings can be made from many types of numbers and objects. Simple examples are whole numbers and fractions. Rings can also include more complex things like polynomials (expressions with variables) or matrices (square grids of numbers).

Direct product

When you have two rings, you can combine them into a new ring called the direct product. This new ring pairs up elements from each original ring. For example, if you have rings R and S, their direct product has pairs like (r, s), where r is from R and s is from S. You add and multiply these pairs by doing the operation separately in each part:

• (r₁, s₁) + (r₂, s₂) = (r₁ + r₂, s₁ + s₂)
• (r₁, s₁) × (r₂, s₂) = (r₁ × r₂, s₁ × s₂)

Polynomial ring

A polynomial ring is a ring made by adding a variable to an existing ring. For example, if you start with whole numbers, you can create polynomials like 2x + 3 or x² - 5y + 1. These polynomials follow the usual rules of addition and multiplication.

Matrix ring and endomorphism ring

A matrix ring is a ring made from square grids of numbers, called matrices. You can add and multiply matrices in a special way that follows rules similar to regular multiplication but with some differences. For example, multiplying matrices isn’t always the same if you switch the order.

Limits and colimits of rings

Sometimes, mathematicians study sequences of rings where each ring fits inside the next one. The union of all these rings (called a colimit) is a new ring that contains all of them. For example, a polynomial ring with infinitely many variables can be built this way.

Localization

Localization is a way to modify a ring by making certain elements behave like the number 1. For example, in the whole numbers, you can create fractions by allowing division by any number except zero. This process helps mathematicians study special properties of rings.

Completion

The completion of a ring at an ideal (a special subset) creates a new ring that fills in "gaps" in the original one. This is useful in studying numbers like p-adic integers, which are like whole numbers but with extra information about primes.

Rings with generators and relations

The most general way to build a ring is to start with some basic building blocks (called generators) and then impose rules (called relations) on how they can combine. For example, you might start with symbols x and y and say that xy must equal yx. This creates a new ring that follows those rules.

Special kinds of rings

A ring is a set with two operations, addition and multiplication, that work like adding and multiplying whole numbers. In a ring, multiplication does not always give the same answer if you switch the order of the numbers.

Rings can be made from numbers, like whole numbers or complex numbers, but they can also be made from other math ideas, such as formulas, grids of numbers, functions, and special number sequences. There are special types of rings with extra rules. For example, a domain is a ring with no zero-divisors, and an integral domain is a domain where multiplication order does not matter. A division ring is a ring where every number (except zero) can be used to divide another number evenly, and a field is a division ring where multiplication order does not matter.

Rings with extra structure

A ring is like a group of numbers where you can add and multiply. There are also special types of rings with extra rules. For example, an associative algebra is a ring that also works like a space with scalars, such as matrices over real numbers. A topological ring is a ring where the elements have a special organization that keeps addition and multiplication smooth, like matrices with certain types of organization. A λ-ring is a special kind of ring used in advanced math. Finally, a totally ordered ring is a ring where the numbers can be compared in a way that fits with addition and multiplication.

Some examples of the ubiquity of rings

Many different kinds of mathematical objects can be studied using rings. For example, to any topological space we can connect a special kind of ring called its integral cohomology ring. This helps us understand shapes like spheres and tori better.

Rings also appear in other areas. To any group, we can connect something called its Burnside ring, which helps describe how the group can act on sets. Similarly, every algebraic variety has a function field, and every simplicial complex has a face ring that shows us information about its shape. These examples show just how useful rings are in many parts of mathematics.

Category-theoretic description

See also: Category of rings

Every ring is a special kind of structure called a monoid in Ab, the category of abelian groups. This means we can study rings using ideas from category theory. An R-module is like a generalization of a vector space, but we work over a ring instead of a field.

For any abelian group A, we can look at its endomorphism ring. This is the set of all ways to map A to itself while keeping the group operation the same. This endomorphism ring has properties that make it a ring. Every ring can be seen as the endomorphism ring of some abelian group. This shows how rings are connected to the study of group mappings.

Generalization

Algebraists study structures that are like rings but with some rules changed or removed. These structures still have addition and multiplication.

A rng is similar to a ring but does not need a special number called a "multiplicative identity." A nonassociative ring is another type that does not follow all the usual rules, such as needing multiplication to be associative. An example of this is a Lie algebra. A semiring is another generalization where the rules for addition are changed, and a special rule about zero and multiplication is added. Examples of semirings include the non-negative integers { 0 , 1 , 2 , … } with ordinary addition and multiplication, and the tropical semiring.

Other ring-like objects

In mathematics, there are special ways to think about rings in different areas.

In categories, a ring object is a special object that behaves like a ring, with rules for addition and multiplication.

In algebraic geometry, a ring scheme is a type of ring object used to study structures over a base scheme.

In algebraic topology, a ring spectrum is a special kind of spectrum with rules that make it behave like a ring, used to study shapes and spaces.