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, $\left(G, \cdot \right)$, a ring can be conceptualized as an ordered triple $\left(R,+,\cdot \right)$.

A set with addition and multiplication, $\left(R,+,\cdot \right)$, 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 $a,b,c\in R$, $\left(a+b\right)+c=a+\left(b+c\right)$ and $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$;
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 $a,b,c\in R$, $a\cdot \left(b+c\right)=a\cdot b+a\cdot c$ and $\left(b+c\right)\cdot a=b\cdot a+c\cdot a$;
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, $a + bc + d = a + \left(bc\right) + d$.

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

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:

• $\left(R,+\right)$ is an abelian group;
• $0\cdot a=0$, for all $a \in R$;
• $a\left(-b\right) = \left(-a\right)b = -\left(ab\right)$, for all $a,b\in R$;
• $\left(-a\right)\left(-b\right)=ab$, for all $a,b\in R$;
• If $R$ has unity $1$:
  • $\left(-1\right)\cdot a=-a$, for all $a \in R$;
  • $\left(-1\right)\cdot \left(-1\right)=1$;
• 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