MHB Is {0} Considered the Smallest Integral Domain?

[Kiwi1]

What is the smallest (most trivial) integral domain?

{0,1} is an integral domain but is {0} an integral domain with unity = 0 and zero = 0?

If {0} is not an integral domain then why not?

 

[Opalg]

What is the smallest (most trivial) integral domain?

{0,1} is an integral domain but is {0} an integral domain with unity = 0 and zero = 0?

If {0} is not an integral domain then why not?

The usual definition of an integral domain (as given here for example) is that it is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. The first of the three occurrences of the word nonzero in that definition is designed to exclude the case of a ring consisting of the single element 0.