To a certain degree, the notion of a ring was invented in an attempt to put the algebraic properties of the integers into an abstract setting. A ring is not the appropriate abstraction of the integers, however, for too much is lost in the process. Besides the two obvious properties of commutativity and existence of a unity, there is one other essential feature of the integers that rings in general do not enjoy—the cancellation property. In this chapter, we introduce integral domains—a particular class of rings that have all three of these properties. Integral domains play a prominent role in number theory and algebraic geometry (Gallian, 2009). # Definition(s) > [!NOTE] Definition (Integral Domain) > An integral domain $(D,+,\times)$ is a non-[[Zero Ring|zero]] [[Commutative Ring|commutative ring with unity]] in which they are no [[Proper Zero Divisor|proper zero divisors]]. > [!Example] Examples > Note that [[Subrings of a field is are integral domains|a field is an integral domain]]. > >The integers form an integral domain by [[Ring of Integers has no Proper Zero Divisors|(Z,+,x) has no proper zero divisors]]. > >[[Ring of Polynomial Forms]] over integral domains [[Ring of Polynomial Forms over Integral Domain is Integral Domain|are integral domains]]. # Properties **Finite Domains**: A [[Finite Integral Domains are Fields|finite integral domain forms a field]] (a finite field is also called a Galois field). **Cancellation**: ...'