Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides

A ring is a set $R$ carrying two binary operations (Binary operation on a set; associativity, commutativity, and a subset closed under the operation), addition $+$ and multiplication $\cdot$, and two distinguished elements $0$ and $1$ of $R$, such that

• (R1) $(R, +, 0)$ is an abelian group (Group and abelian group): addition is associative and commutative, $0 + x = x = x + 0$ for every $x \in R$, and every $x \in R$ has an additive inverse;
• (R2) $(R, \cdot, 1)$ is a monoid (Semigroup and monoid): multiplication is associative and $1 \cdot x = x = x \cdot 1$ for every $x \in R$;
• (R3) multiplication distributes over addition on both sides: for all $x, y, z \in R$,

$x \cdot (y + z) \;=\; x \cdot y + x \cdot z, \qquad (y + z) \cdot x \;=\; y \cdot x + z \cdot x .$

We write $xy$ for $x \cdot y$, call $0$ the zero and $1$ the identity (or unity) of $R$, write $-x$ for the additive inverse of $x$, and abbreviate $x - y := x + (-y)$.

Why "the" zero, "the" identity and "the" additive inverse are legitimate. Each of $0$ and $1$ is a two-sided identity for its own operation, and a binary operation has at most one two-sided identity (A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity, Left identity, right identity, and two-sided identity for a binary operation), so each is determined by the operation and the notation is single-valued. An additive inverse of $x$ is an inverse in the monoid $(R,+,0)$ (Left inverse, right inverse, and invertible element of a monoid), and in a monoid an element has at most one inverse (In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided); so $-x$ names one element, and a single equation $y + x = 0$ already forces $y = -x$.

Convention: a ring has an identity. Axiom (R2) asks for a monoid, so the existence of $1$ is part of the definition and is not an extra hypothesis to be carried around. This is the convention used throughout this library. A structure satisfying (R1), (R3) and the associativity half of (R2), but not required to have a multiplicative identity, is called a non-unital ring (also rng); it is not called a ring here, and where such a structure occurs it is named as a non-unital ring in as many words. The distinction has content: the companion page exhibits a subset of $\mathbb{Z}$ that is closed under addition, additive inverses and multiplication and is not a subring, precisely because it misses the identity.

Nothing above requires $1 \ne 0$. A ring in which $1 = 0$ has exactly one element; the companion page records it as the zero ring. Definitions that need $1 \ne 0$ — integral domain, division ring, field — say so explicitly.

• Both distributive laws are stated, and neither is redundant here. Multiplication is not assumed commutative, so $(y+z)x = yx + zx$ does not follow from $x(y+z) = xy + xz$. The published Field states only the left form, in a setting where multiplication is commutative; a ring is not that setting. The quaternions (The quaternions $\mathbb{H}$: real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$), constructed on this page, are a ring whose multiplication is not commutative, so for them the two laws are genuinely two statements and both are checked.

• A ring packages two structures the library already has. Read (R1) alone and a ring is an abelian group; read (R2) alone and it is a monoid. Every theorem proved about groups and about monoids therefore applies to a ring without restatement, and this page uses that repeatedly rather than reproving cancellation, uniqueness of inverses, or the behaviour of finite products.

• Notation for repeated addition. For $m \in \mathbb{Z}$ and $a \in R$ the expression $ma$ means the integer power of $a$ in the additive group $(R,+,0)$, written additively as in Powers $g^{n}$: natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$. It is not a product formed with the ring multiplication: $m$ is an external integer scalar, even when the underlying set of $R$ happens to contain integers. The arithmetic of these multiples is Integer multiples in a ring: $(m + n)a = ma + na$, $m(a + b) = ma + mb$, $(ma)b = m(ab) = a(mb)$ and $(ma)(nb) = (mn)(ab)$ for all $m, n \in \mathbb{Z}$ and $a, b \in R$.