PRINCIPAL IDEAL DOMAIN Meaning and
Definition
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A principal ideal domain (PID) is a type of mathematical structure in abstract algebra. It is a commutative ring in which every ideal is a principal ideal. In other words, every ideal in a principal ideal domain can be generated by a single element of the ring.
Formally, let R be a commutative ring with a multiplicative identity 1. R is a principal ideal domain if every ideal I of R can be written as I = (a) for some element a in R, where (a) denotes the ideal generated by a. This means that for every element b in I, there exists an element c in R such that b = ac.
Principal ideal domains are significant in algebraic number theory and algebraic geometry, as they provide a rich class of rings that possess many useful properties. For instance, every principal ideal domain is a unique factorization domain, meaning that every nonzero non-unit element can be factored into a unique product of prime elements up to order and associates.
Furthermore, principal ideal domains are integral domains, which means they have no zero divisors. This property makes them well-suited for solving equations and studying divisibility properties.
The concept of principal ideal domains serves as a powerful tool in many areas of mathematics, providing a framework for studying factorization, divisibility, and algebraic structures.
