A commutative unital ring that is not an integral domain Prove or disprove: a commutative unital ring that is not an integral domain may be
contained in a field.
Looking at the definitions, I thought that an integral domain was a ring with unity.
 A: An integral domain is more than a ring with unity. An integral domain must be commutative and not have divisors of zero. A divisor of zero of a ring $R$ is an element $r\in R\setminus\{0\}$ for which there exists another element $s\in R\setminus\{0\}$ such that $rs=0$.
Try proving that a field cannot have divisors of zero. This will lead you to the conclusion that a commutative unital ring that has divisors of zero cannot be contained in a field as a subring. It may of course be contained as a subset.
A: An integral domain is a special type of commutative, unital ring - that is, a commutative, unital ring is only an integral domain if it satisfies a certain extra property. 
Specifically, a commutative, unital ring $R$ is called an integral domain when the following statement is true:
$$\text{For all }a,b\in R,\; ab=0\implies a=0\,\text{ or }\,b=0.$$
Therefore, a commutative, unital ring $R$ is not an integral domain when the negation of the above statement is true:
$$\text{There exist }a,b\in R\text{ such that }ab=0,\text{ but }a\neq0\text{ and }b\neq0.$$
Suppose that a non-integral-domain $R$ were a subring of a field $F$. What do you know about nonzero elements of a field? Do you see how can you apply that to reach a contradiction?
