What is the importance of integral domains? What is the importance of integral domains?
In abstract algebra
Thanks for your help
 A: @Cameron Buie is correct, if a bit brief.  Integral domains eliminate two "undesirable" properties from a general ring: non-commutativity, and zero divisors.  I think the latter property is more important, because if we eliminate zero divisors, then we gain the cancellation rule: if $ca = cb$, then $a = b$.  We can't say this if we have zero divisors: i.e. $a,b$ are zero divisors if $ab=0$ but $a,b\ne0$. Zero divisors occur in non-trivial situations: e.g. matrix rings, modulo integer arithmetic for non-prime modulus, etc.  Integral domains then go on to have other nice properties (e.g. a quotient field), but on an intuitive level, commutativity and cancellation are very nice properties on their own.
A: It is a very useful abstraction.  It rests on a huge field of examples.  One thing that is interesting is that if $R$ is an i.d., then $R[X]$is an i.d. too.  
A: Definition:
An Integral domain is a commutative ring( with unity) $(R,+,\cdot)$ for which the following property holds
If $a,b \in R$ such that $a,b \neq 0$, then $ab\neq 0$. 
To see some motivation behind the definition of an Integral domain, notice that when we were little kids, while solving polynomial equations in $\mathbb Z$, after factoring a polynomial, we make the jump from 

$$x(x-2)(x-3)=0$$ 

$$\text{to}$$ 

$$x=0 \text{ or } x=2 \text{ or } x=3$$

This is a convenient property to have, since it almost always makes it easy to find ALL solutions of a polynomial equation.And this property holds for all elements if and only if the ring is an integral domain. This is quite easy to verify.
Another motivation is the following question: 

Given a ring $R$, when does there exist a field $F$ and an injective ring homomorphism $\phi:R \to F$? 

Informally, we are asking when a given ring $R$ can be described as a subring of a field. The answer turns out that this is possible if and only if $R$ is an integral domain.
One direction of this is easy. Suppose that $R$ can be embedded into a field as above. For any two elements $a,b\neq 0 \in R$, if $ab=0$, then $\phi(ab)=0$ too. Since $\phi$ is a homomorphism, this means that $\phi(a)\phi(b)=0$, Since $\phi$ is injective, neither $\phi(a)$ or $\phi(b)$(Both elements of $F$) are zero. Since $F$ is a field, this means they must be invertible. Multiplying on the left by $(\phi(a))^{-1}$, we see that $\phi(b)=0$, a contradiction. So, $ab \neq 0$. So, $R$ is an integral domain.
For the other direction, see Field of Fractions. This is a construction which not only constructs a field containing our Integral Domain, but also constructs the smallest such one. For example, the Field of Fractions of $\mathbb Z$ is $\mathbb Q$, the field of Rational numbers. This might clarify which the construction is called the field of fractions, since $\mathbb Q$ is what we usual think of as fractions.
Thus, the notion of an abstract Integral domain is closely modeled after the Integers.
