Ring

Definition

A Ring is an algebraic structure consisting of a set equipped with two operations: addition and multiplication. If the set $R$ has well-defined addition ($+$) and multiplication ($\cdot$) operations that satisfy the following properties, then $(R,+,\cdot )$ is called a ring.

1. $(R,+)$ forms a Commutative Group:

   • Associativity: $a+(b+c)=(a+b)+c$
   • Commutativity: $a+b=b+a$
   • Existence of an identity element: $a+0_R=a$
   • Existence of inverse elements: For each $a\in R$, there exists $x\in R$ such that $a+x=0_R$

2. $(R,\cdot )$ forms a Monoid:

   • Associativity: $a\cdot (b\cdot c)=(a\cdot b)\cdot c$
   • Existence of an identity element: $a\cdot 1_R=a$

3. Distributive laws hold:

   • $(a+b)\cdot c=a\cdot c+b\cdot c$
   • $c\cdot (a+b)=c\cdot a+c\cdot b$

Types

• Commutative Ring
• Unit Ring
• Division Ring
• Quotient Ring
• Pseudoring