Ring (mathematics)

In mathematics, a ring is a special kind of algebraic structure that has two main operations, called addition and multiplication. These operations behave much like the addition and multiplication you know from regular numbers, but with one big difference: in a ring, multiplying numbers doesn't always give the same result no matter what order you do it in. Ring elements can be numbers like integers or complex numbers, but they can also be other things, like polynomials, square matrices, functions, and power series.

A ring is defined more formally as a set where addition forms 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 follows the same order no matter which numbers you multiply, it is called a commutative ring. Commutative rings are very important in mathematics because they help us understand many areas 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 with n ≥ 2. The idea of rings was developed between the 1870s and 1920s by mathematicians such as Richard Dedekind, David Hilbert, Abraham Fraenkel, and Emmy Noether. Today, rings are a fundamental part of many areas of mathematics.

Definition

A ring is a special set of numbers or objects with two operations, called addition and multiplication. These operations work much like regular addition and multiplication, but there are a few important rules.

Addition in a ring must follow the same rules as normal addition: it is associative (the order of operations doesn’t change the result), commutative (the order of the numbers doesn’t change the result), and there must be an additive identity (a number that 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. Additionally, multiplication must distribute over addition. This means that 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, which includes numbers like ..., -5, -4, -3, ..., 0, 1, 2, 3, .... These numbers follow special rules when we add or multiply them, similar to 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, though he didn’t use that word.

Later, in 1892, David Hilbert used the term “number ring” to describe certain number patterns that repeat in a special way. Around 1915, Abraham Fraenkel created the first strict rules for what a ring should be, but his rules were tighter than what we use today. In 1921, Emmy Noether improved these rules and laid the groundwork for how we understand rings today.

Over time, mathematicians have debated whether rings should always include a special number called “1”. Some books say yes, others say no, and writers usually state which rule they are using.

Basic examples

See also: Associative algebra § Examples

Commutative rings

The simplest example of a ring is the set of integers, where we use the usual addition and multiplication. Other examples include the rational numbers, real numbers, and complex numbers, which are special types of rings called fields. We can also create rings using polynomials, which are expressions like (x^2 + 2x + 1), where the coefficients come from another ring. For example, polynomials with integer coefficients form a ring. Similarly, functions that take real numbers as inputs and give real numbers as outputs can also form a ring under certain conditions.

Noncommutative rings

Not all rings have multiplication that works the same way as with numbers. For example, square matrices — grids of numbers — can form a ring where the multiplication is done by a special matrix multiplication rule. If the matrices are bigger than just one by one, their multiplication does not follow the same order as regular numbers, making these rings noncommutative. Another example is the set of all ways to change or map elements of certain mathematical structures, which can also form rings with their own special 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 because they lack certain properties, such as having numbers that can “undo” addition (like negative numbers for the integers). These natural numbers can be expanded to include negative numbers, forming the ring of integers. There are also special cases where multiplication is defined in a non-standard way, leading to structures that are similar to rings but not quite rings themselves.

Basic concepts

A ring is a special kind of mathematical structure with two operations, like adding and multiplying numbers. These operations behave somewhat like they do with integers, but there’s a twist: multiplication doesn’t have to commute. This means that for some rings, swapping the order of multiplication can give a different 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 instead of using numbers to multiply, we use elements from a ring. A module is a special kind of group that follows certain rules when we multiply its elements by elements of the ring. These rules help us understand how the elements interact with each other.

One big difference between modules and vector spaces is that modules don't always have a "size" like vector spaces do. This makes studying modules more complex. For example, not all modules have a basis — a set of elements that can be used to build every other element in the module. Modules help us explore 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 kind of math structure that has two operations: addition and multiplication. These operations work a lot like adding and multiplying whole numbers, but with some differences. For example, multiplication in a ring doesn’t always work the same way as regular multiplication — the order might not matter.

Rings can be made from many different kinds of numbers and objects. Simple examples include whole numbers and fractions, but rings can also include more complex things like polynomials (expressions with variables and powers) 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 behave like the addition and multiplication of integers. However, in a ring, multiplication does not have to be commutative, meaning the order in which you multiply can change the result.

Rings can be made from numbers, like integers or complex numbers, but they can also be made from other mathematical objects, such as polynomials, matrices, functions, and power series. There are special types of rings with additional properties. For example, a domain is a ring with no zero-divisors, and an integral domain is a commutative domain. A division ring is a ring where every non-zero element has a multiplicative inverse, and a field is a commutative division ring. Semisimple rings are rings that are simple as modules over themselves, and they have special properties that make them useful in various areas of mathematics.

Rings with extra structure

A ring is like a group of numbers where you can add and multiply, but 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, like whole numbers with extra operations based on combinations. 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 can be thought of as 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, where instead of working over a field, we work over a ring.

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

Generalization

Algebraists have created structures more general than rings by changing or removing some of the rules that rings follow. These new structures still have addition and multiplication, but they might not follow all the same rules.

A rng is almost the same as a ring, but it does not require a special number called a "multiplicative identity." A nonassociative ring is another type of structure that does not follow all the usual rules, such as the need for 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.