Is every integral domain a field?
The question of whether every integral domain is a field has intrigued mathematicians for centuries. An integral domain is a commutative ring with no zero divisors, while a field is a commutative ring with unity in which every nonzero element has a multiplicative inverse. At first glance, it may seem intuitive that every integral domain should be a field, but this is not the case. In this article, we will explore the relationship between integral domains and fields, and provide a counterexample to demonstrate that not every integral domain is a field.
An integral domain is a commutative ring with no zero divisors, meaning that if a and b are elements of the ring and ab = 0, then either a = 0 or b = 0. A field, on the other hand, is a commutative ring with unity in which every nonzero element has a multiplicative inverse. This implies that in a field, for every element a, there exists an element b such that ab = 1.
The question of whether every integral domain is a field can be answered by considering the definition of a field. If an integral domain has a multiplicative identity (1), then every nonzero element in the domain must have a multiplicative inverse. However, this is not always the case. Consider the ring of integers, denoted by Z. Z is an integral domain because it has no zero divisors, but it is not a field because not every nonzero element has a multiplicative inverse. For example, the element 2 in Z has no multiplicative inverse in Z, as there is no integer b such that 2b = 1.
Another counterexample to the statement “every integral domain is a field” is the ring of polynomials with coefficients in a field, denoted by F[x]. F[x] is an integral domain because it has no zero divisors, but it is not a field. In F[x], the polynomial x has no multiplicative inverse, as there is no polynomial p(x) such that x p(x) = 1.
The reason why not every integral domain is a field lies in the fact that an integral domain may not have a multiplicative identity. In the case of the ring of integers, the multiplicative identity is 1, but in the ring of polynomials, there is no multiplicative identity. This absence of a multiplicative identity is what prevents an integral domain from being a field.
In conclusion, the statement “every integral domain is a field” is false. While integral domains and fields share some properties, such as being commutative and having no zero divisors, the presence of a multiplicative identity is what distinguishes a field from an integral domain. By providing counterexamples such as the ring of integers and the ring of polynomials, we have demonstrated that not every integral domain is a field. This highlights the importance of understanding the fundamental differences between these two types of rings in the study of abstract algebra.
