Summary

Ring theory

Definition

A ring is an ordered tuple $(R,+,\cdot )$ where $R$ is a set and $+$ and $\cdot$ are binary operations on $R$ such that:

• $(R,+)$ is an Abelian group.
• $\cdot$ is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for all $a,b,c\in R$.
• The distributive laws hold: $a\cdot (b+c)=a\cdot b+a\cdot c$ and $(a+b)\cdot c=a\cdot c+b\cdot c$ for all $a,b,c\in R$.

Note

• A ring is a not necessarily commutative group. Hence, the multiplication operation may not be commutative.

Types of rings

Commutative ring

Definition

A commutative ring is a ring where the multiplication operation is commutative, i.e., $a\cdot b=b\cdot a$ for all $a,b\in R$.

Examples

• The set of integers $\mathbb{Z}$ under addition and multiplication is a commutative ring.
• The set of rational numbers $\mathbb{Q}$ under addition and multiplication is a commutative ring.

Non-commutative ring

Definition

A non-commutative ring is a ring where the multiplication operation is not commutative, i.e., $a\cdot b\ne b\cdot a$ for some $a,b\in R$.

Examples

• The set of $n\times n$ matrices over a field is a non-commutative ring.
• The set of quaternions is a non-commutative ring.

Finite ring

Definition

A finite ring is a ring with a finite number of elements.

Examples

• The ring of integers modulo $n$ is a finite ring.
• The ring of $n\times n$ matrices over a field is a finite ring.

Infinite ring

Definition

An infinite ring is a ring with an infinite number of elements.

Examples

• The ring of integers $\mathbb{Z}$ is an infinite ring.
• The ring of polynomials over a field is an infinite ring.

Ring with identity

Definition

A ring with identity is a ring that has a multiplicative identity element, denoted by $1$.

Examples

• The set of integers $\mathbb{Z}$ under addition and multiplication is a ring with identity.
• The set of real numbers $\mathbb{R}$ under addition and multiplication is a ring with identity.

Ring without identity (Rng)

Definition

A ring without identity is a ring that does not have a multiplicative identity element.

Examples

• The set of even integers under addition and multiplication is a ring without identity.
• The set of $n\times n$ matrices over a field is a ring without identity.

Polynomial ring

Definition

Let a ordered tuple $(R,+,\cdot )$ be a ring. A polynomial ring is a ring formed by polynomials over the ring $R$.

A polynomial, denoted by $f(x)$, is a formal expression of the form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x+a_0$ where $a_i\in R$ and $n$ is a non-negative integer. The set of all polynomials in the variable $x$ with coefficients in the ring $R$ forms a polynomial ring denoted by $R[x]$.

Examples

• The ring of polynomials with coefficients in $\mathbb{Z}$ is a polynomial ring.
• The ring of polynomials with coefficients in a $n\times n$ matrix ring is a polynomial ring.