Ring

A ring is an algebraic structure comprised of a set paired with two operations on the set, which are designated as addition $(+)$ and multiplication $(\cdot)$ . As a group can be conceptualized as an ordered pair of a set and an operation, $(G,\cdot)$ , a ring can be conceptualized as an ordered triple $(R, +, \cdot)$ .

A set with addition and multiplication, $(R, +, \cdot)$ , is a field if and only if it satisfies the following properties:

1. Commutativity of addition — For all $a,b\in R$ , $a + b = b + a$
2. Associativity of both addition and multiplication — For all $a,b,c\in R$ , $(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot c=a\cdot(b\cdot c)$
3. Additive Identity — There exists a "zero" element, $0\in R$ , called an additive identity, such that $a+0=a=0+a$ , for all $a \in R$
4. Additive Inverses — For each $a \in R$ , there exists a $b\in R$, called an additive inverse of $a$ , such that $a+b=0$
5. Distributive property — For all $a,b,c\in R$ , $a\cdot(b+c)=a\cdot b+a\cdot c$ and $(b+c)\cdot a=b\cdot a+c\cdot a$
6. Closure of addition and multiplication — For all $a,b\in R$ , $a+b\in R$ and $a\cdot b\in R$

A prototype of these kind of structures in the ring of the integers, $(\Z,+,\cdot)$ .

We will often abbreviate the multiplication of two elements, $a\cdot b$ , by juxtaposition of the elements, $ab$ . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, $a+bc+d=a+(bc)+d$ .

We can also denote $-a$ as the additive inverse of any $a \in R$ . Furthermore, we can define an operation, called subtraction by $a-b=a+(-b)$ .

Optional Properties

Optionally, a ring $R$ may have additional properties:

• We define $R$ to be a commutative ring if the multiplication is commutative: $a\cdot b=b\cdot a$ for all $a,b\in R$
• We define $R$ to be a ring with unity if there exists a multiplicative identity $1\in R$ : $1\cdot a=a=a\cdot1$ for all $a \in R$
  • Furthermore, a commutative ring with unity $R$ is a field if every element except 0 has a multiplicative inverse: For each non-zero $a \in R$ , there exists a $b\in R$ such that $a\cdot b=b\cdot a=1$
• We define $R$ to be an integral domain if it is commutative, has unity, and the zero product rule holds: $a\cdot b=0$ implies either $a = 0$ or $b=0$ for all $a,b\in R$ .

Important Results

From the given axioms for a ring $R$ , it can be deduced that:

• $(R, +)$ is an abelian group
• $0\cdot a=0$ , for all $a \in R$
• $a(-b)=(-a)b=-(ab)$ , for all $a,b\in R$
• $(-a)(-b)=ab$ , for all $a,b\in R$
• If $R$ has unity $1$
  • $(-1)\cdot a=-a$ , for all $a \in R$
  • $(-1)\cdot(-1)=1$
• Multiplication distributes over subtraction.

Substructures, morphisms and quotients

• subring
• ideal
• prime ideal
• maximal ideal
• field
• radical
• nilradical
• ring homomorphism
• quotient ring
• fundamental theorems

Examples and results

• integers
• chinese remainder theorem
• gaussian integers
• polynomial ring
• integral domain
• euclidean ring
• unique factorization domain
• non commutative ring