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?
An integral domain is a mathematical structure that consists of a set of elements, a set of operations, and a set of axioms that govern the behavior of the elements and operations. In an integral domain, every element has a unique multiplicative inverse, and the operations of addition and multiplication satisfy the distributive, associative, and commutative properties.
The smallest integral domain is the field of integers modulo 2, denoted as Z2. It consists of two elements, 0 and 1, and the operations of addition and multiplication are defined modulo 2, meaning that the results are either 0 or 1.
The smallest integral domain, Z2, is important because it is the building block for constructing larger integral domains. It is also used in many applications, such as error-correcting codes in computer science and cryptography.
The smallest integral domain, Z2, has the properties of an integral domain, including the existence of a multiplicative identity, the existence of unique multiplicative inverses, and the distributive, associative, and commutative properties of addition and multiplication. Additionally, it is a finite field, meaning that it has a finite number of elements.
The smallest integral domain, Z2, is unique in that it only has two elements and is the only integral domain that is also a finite field. Other integral domains may have an infinite number of elements and may have additional properties, such as being a Euclidean domain or a principal ideal domain.