An Integral Domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
Every nonzero element ‘a’ has the Cancellation Property (ab=ac –> b=c).
We say an Integral Domain has a Multiplicative Identity. Some authors do not require this.
We will follow the convention of saying that an Integral Domain is commutative, and if it wasn’t commutative, then we would just call it a Domain.
Every Subring of a Field us an Integral Domain.
Let R be an Integral Domain with elements a,b and also an element x in R such that
ax=b
- a divides b
- a is a divisor of b
- b is a multiple of a
Appendix A
After reading the first paragraph it would be fair to ask, can we multiply two nonzero numbers and get a zero?
Yes, using mod math. The additions below are mod 3 additions on the set {0,1,2}
- 2+0=2
- 2+1=0
- 2+2=1
Both 2 and 1 are nonzero and their mod 3 addition results in 0.
Rob Sterling 