Metric_Space said:
A prime ideal is a subset of a ring that shares some properties with prime numbers in the integers. Specifically, it is an ideal that is not equal to the whole ring and has the property that if the product of two elements is in the ideal, then at least one of the elements must be in the ideal.
A regular ideal is simply a subset of a ring that is closed under addition and multiplication by elements of the ring. A prime ideal has the additional property that it is not equal to the whole ring and has the restriction on products described in the previous answer.
Prime ideals play a crucial role in the study of abstract algebra. They help to classify rings and provide a way to understand their structure. They are also important in the study of fields, as the set of prime ideals of a ring corresponds to the set of irreducible polynomials in a field extension.
In a ring with unique factorization, every element can be written as a product of irreducible elements. Prime ideals play a key role in this process, as they correspond to the irreducible elements of the ring. By identifying prime ideals, we can determine the prime factorization of elements in the ring.
No, prime ideals are not unique. Just as there can be multiple prime numbers that divide a given integer, there can be multiple prime ideals in a given ring. However, in certain cases, such as in a principal ideal domain, prime ideals can be unique.