UFD (Unique Factorization Domain)

A Unique Factorization Domain (UFD) is a type of mathematical system where elements, like numbers or algebraic objects, can be broken down into prime building blocks uniquely, much like how integers can be factored into prime numbers (e.g., 12 = 2 × 2 × 3). In a UFD, this factorization is consistent—every element has only one way to be expressed as a product of primes, apart from order. This property simplifies understanding and working with these systems, ensuring that prime factors are fundamental and uniquely determine the structure, aiding in solving equations and proving properties within algebraic contexts.