The discussion centers on the classification of the set {0} as an integral domain. It is established that {0} cannot be considered an integral domain because it fails to meet the definition of a nonzero commutative ring, specifically due to the absence of nonzero elements. The set {0,1} is confirmed as an integral domain, as it contains nonzero elements and satisfies the necessary properties. The key takeaway is that an integral domain must have at least one nonzero element to fulfill its definition.
PREREQUISITESMathematics students, educators in abstract algebra, and anyone interested in the foundational concepts of ring theory and integral domains.
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.Kiwi said: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?