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Ring

Definition

A ring is an algebraic structure (A,+,⋅)(A,+,\cdot) such that:

  1. (A,+)(A,+) is a commutative group;
  2. (A,⋅)(A,\cdot) is a semigroup;
  3. multiplication distributes over addition:
a⋅(b+c)=a⋅b+a⋅c,(a+b)⋅c=a⋅c+b⋅c.a\cdot(b+c)=a\cdot b+a\cdot c,\qquad (a+b)\cdot c=a\cdot c+b\cdot c.

Interpretation

A ring is a setting with two operations, addition and multiplication, that behave like those of the integers: one can add, subtract and multiply, and brackets can be expanded in the usual way. Division is not part of the definition.

Each part of the definition has its own role. Since (A,+)(A,+) is an Abelian group, addition is associative and commutative, has a zero element 00, and every element has an additive inverse −a-a, so subtraction a−b=a+(−b)a-b=a+(-b) is always available. Multiplication is only asked to form a semigroup: it is associative, but it need not be commutative and need not have an identity element. The two distributive laws link the operations, and both are needed because a⋅ba\cdot b and b⋅ab\cdot a may differ. Familiar rules follow from them, for example a⋅0=0a\cdot0=0: from a⋅0=a⋅(0+0)=a⋅0+a⋅0a\cdot0=a\cdot(0+0)=a\cdot0+a\cdot0, adding −(a⋅0)-(a\cdot0) on both sides gives 0=a⋅00=a\cdot0.

The comparison with a field shows what is missing. (Z,+,⋅)(\mathbb{Z},+,\cdot) is a ring, but 22 has no multiplicative inverse in Z\mathbb{Z}, so 2x=12x=1 has no integer solution. The polynomials with real coefficients also form a ring: they can be added, subtracted and multiplied, but the quotient of two polynomials is usually not a polynomial, which is why polynomial long division works with a remainder. The square matrices of a fixed size n≥2n\ge2, which appear later in this course, form a ring whose multiplication is not commutative.

Key Results

  • If multiplication is commutative, the ring is a commutative ring.
  • If multiplication has an identity element, the ring is a ring with unity.

Examples

  • (Z,+,⋅)(\mathbb{Z},+,\cdot) is a commutative ring with unity.
  • (R,+,⋅)(\mathbb{R},+,\cdot) is a commutative ring with unity: addition has identity 00 and additive inverses, multiplication is associative with identity 11, and multiplication distributes over addition.

Prerequisites

Sources

Symbols: integers, real numbers

Last updated: 22 October 2026 19:04