Ring (mathematics) facts for kids

Richard Dedekind, one of the founders of ring theory.

This article is about a mathematical concept. For the piece of jewellery, see ring.

In mathematics, a ring is a special kind of algebraic structure. Think of it as a collection of things (called a set) that you can add and multiply together. These two actions, addition (+) and multiplication (•), must follow specific rules to make the collection a "ring."

Mathematicians use the word "ring" because a mathematician named David Hilbert used the German word Zahlring, which means "number ring." Famous examples of rings include integers (whole numbers), rational numbers (fractions), real numbers (like pi), and complex numbers. There are also many other types of rings, but they all follow the same special rules.

What Are the Rules for a Ring?

For a set of numbers and two operations (addition and multiplication) to be called a ring, they must follow a few important rules. These rules are called ring axioms.

• Closure: When you add or multiply any two elements from the ring, the answer must also be part of that same ring.
  • For example, if you have numbers in a ring, and you add two of them, the sum must also be a number in that ring.
  • The same goes for multiplication: multiply two numbers from the ring, and the product must stay in the ring.

• Additive Identity: There's a special element in every ring called the "additive identity." When you add this element to any other element, the other element doesn't change.
  • This special element is usually called "zero" (0). So, 0 + a = a + 0 = a.

• Associativity of Addition: When you add three or more elements, the way you group them doesn't change the final answer.
  • For example, (a + b) + c gives the same result as a + (b + c).

• Commutativity of Addition: When you add two elements, the order doesn't matter.
  • For example, a + b gives the same result as b + a.

• Additive Inverse: Every element in a ring has a "partner" element. When you add an element to its partner, the result is the additive identity (zero).
  • For example, for any number 'a', there's a '-a' such that a + (-a) = 0.

• Associativity of Multiplication: Just like with addition, when you multiply three or more elements, the way you group them doesn't change the final answer.
  • For example, (a • b) • c gives the same result as a • (b • c).

• Distribution: This rule shows how multiplication and addition work together.
  • It means that a • (b + c) is the same as (a • b) + (a • c).
  • Also, (a + b) • c is the same as (a • c) + (b • c).

Special Kinds of Rings

Some rings have extra properties beyond the basic rules. These rings get special names:

What is a Commutative Ring?

If the order of multiplication doesn't matter (meaning x • y = y • x for all elements x and y), then the ring is called a commutative ring.

What is a Ring with Unity?

If a ring has a special "multiplicative identity" element, it's called a ring with unity. When you multiply any element by this special element, the element doesn't change.

• This special element is usually called "one" (1). So, 1 • a = a • 1 = a.

What is a Division Ring?

In a division ring, every element (except zero) has a "multiplicative inverse." This means for any element 'a', there's another element 'a^-1' that, when multiplied by 'a', gives you the multiplicative identity (one).

• So, a • a^-1 = 1.

What is an Integral Domain?

Sometimes in a ring, you can multiply two things that are not zero and still get zero as the answer. If this is impossible in a specific ring, then it's called an integral domain.

What is a Field?

A ring that has all the properties mentioned above (commutative, with unity, and a division ring) is called a field.

See also

In Spanish: Anillo (matemática) para niños