A ring $R$ is a triple $(X,\oplus ,\otimes )$ where $X$ is a set and $\oplus$ and $\otimes$ are binary operations subject to the ring axioms. We write $x\oplus y$ for the application of $\oplus$ to $x,y\in X$, which must be defined, and similarly for $\otimes$. It is standard to abbreviate $x\otimes y$ as $xy$ when $\otimes$ can be inferred from context. The ten ring axioms (which govern the behavior of $\oplus$ and $\otimes$) are as follows:

1. $X$ must be a commutative group under $\oplus$. That means:

   • $X$ must be closed under $\oplus$.

   • $\oplus$ must be associative.

   • $\oplus$ must be commutative.

   • $\oplus$ must have an identity, which is usually named $0$.

   • Every $x\in X$ must have an inverse $(-x)\in X$ such that $x\oplus (-x)=0$.

2. $X$ must be a monoid under $\otimes$. That means:

   • $X$ must be closed under $\otimes$.

   • $\otimes$ must be associative.

   • $\otimes$ must have an identity, which is usually named $1$.

3. $\otimes$ must distribute over $\oplus$. That means:

   • $a\otimes (x\oplus y)=(a\otimes x)\oplus (a\otimes y)$ for all $a,x,y\in X$.

   • $(x\oplus y)\otimes a=(x\otimes a)\oplus (y\otimes a)$ for all $a,x,y\in X$.

Though the axioms are many, the idea is simple: A ring is a commutative group equipped with an additional operation, under which the ring is a monoid, and the two operations play nice together (the monoid operation distributes over the group operation).

A ring is an algebraic structure. To see how it relates to other algebraic structures, refer to the tree of algebraic structures.

Examples

The integers $Z$ form a ring under addition and multiplication.

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Notation

Given a ring $R=(X,\oplus ,\otimes )$, we say ”$R$ forms a ring under $\oplus$ and $\otimes$.” $X$ is called the underlying set of $R$. $\oplus$ is called the “additive operation,” $0$ is called the “additive identity”, $-x$ is called the “additive inverse” of $x$. $\otimes$ is called the “multiplicative operation,” $1$ is called the “multiplicative identity”, and a ring does not necessarily have multiplicative inverses.

Basic properties

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Interpretations, Visualizations, and Applications

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