MHB Is {0} Considered the Smallest Integral Domain?

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The discussion centers on whether the set {0} can be classified as an integral domain, with participants questioning its status given that it contains only the zero element. It is established that {0} does not qualify as an integral domain because it lacks nonzero elements, which is a requirement in the definition of an integral domain. The set {0,1} is confirmed as an integral domain since it includes nonzero elements and meets the necessary criteria. The definition emphasizes that an integral domain must be a nonzero commutative ring where the product of any two nonzero elements remains nonzero. Ultimately, {0} is excluded from being an integral domain due to its singularity and absence of nonzero elements.
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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?
 
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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?
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.
 
Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

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