# Ring

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:

\(X\) 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 \(x \oplus (-x) = 0\).

\(X\) 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 \(1\).

\(\otimes\) 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 \(\mathbb{Z}\) form a ring under addition and multiplication.

Add more example rings. [work in progress.]

# 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

Add the basic properties of rings. [work in progress.]

# Interpretations, Visualizations, and Applications

Add (links to) interpretations, visualizations, and applications. [work in progress.]

Children:

- Ordered ring
A ring with a total ordering compatible with its ring structure.

- Irreducible element (ring theory)
This is the appropriate abstraction of the concept of “prime number” to general rings.

- Prime element of a ring
Despite the name, “prime” in ring theory refers not to elements which are “multiplicatively irreducible” but to those such that if they divide a product then they divide some term of the product.

- Integral domain
An integral domain is a ring where the only way to express zero as a product is by having zero as one of the terms.

- In a principal ideal domain, "prime" and "irreducible" are the same
Principal ideal domains have a very useful property that we don’t need to distinguish between the informal notion of “prime” (i.e. “irreducible”) and the formal notion.

- Unit (ring theory)
A unit in a ring is just an element with a multiplicative inverse.

- Principal ideal domain
A principal ideal domain is a kind of ring, in which all ideals have a certain nice form.

- Kernel of ring homomorphism
The kernel of a ring homomorphism is the collection of things which that homomorphism sends to 0.

- Ideals are the same thing as kernels of ring homomorphisms
- Euclidean domains are principal ideal domains
A Euclidean domain is one where we may somehow perform the division algorithm; this gives us access to some of the nicest properties of the integers.

- Unique factorisation domain
This is the correct way to abstract from the integers the fact that every integer can be written uniquely as a product of prime numbers.

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