Is Every Irreducible Element of a GCD Domain Prime?

Yes, every irreducible element of a GCD domain is indeed prime. This property is one of the defining characteristics that distinguish GCD domains within the broader field of abstract algebra.

Understanding Irreducible and Prime Elements

To fully appreciate this fact, it's essential to understand the distinction between irreducible and prime elements in an integral domain.

• Irreducible Element: An element p in an integral domain R is irreducible if it is a non-zero, non-unit element that cannot be written as a product of two non-unit elements. In simpler terms, it cannot be factored into "smaller" elements within the domain.
• Prime Element: An element p in an integral domain R is prime if it is a non-zero, non-unit element such that if p divides a product ab, then p must divide a or p must divide b. This concept mirrors the definition of prime numbers in integers.

In any integral domain, every prime element is always irreducible. However, the converse is not universally true. There are integral domains where an irreducible element is not necessarily prime. A common example is the ring of integers with $\sqrt{-5}$ adjoined, $\mathbb{Z}[\sqrt{-5}]$, where, for instance, the element 3 is irreducible but not prime.

GCD Domains: Where Irreducible Implies Prime

In a GCD domain, the concepts of irreducible and prime elements coincide. This means that if an element is irreducible, it must also be prime. This crucial property simplifies factorization theory within these domains.

A GCD domain, or a greatest common divisor domain, is an integral domain where every pair of non-zero elements has a greatest common divisor (GCD). The existence of GCDs for all pairs of elements imposes a strong structure that forces irreducible elements to also be prime.

Key Properties of GCD Domains

GCD domains possess several significant properties that contribute to this equivalence:

• Integrally Closed: A GCD domain is always integrally closed. This means that any element in its field of fractions that is a root of a monic polynomial with coefficients in the domain must already be an element of the domain itself.
• Primal Elements: Every nonzero element in a GCD domain is primal. An element p is primal if, whenever p divides a product ab, and p and a are coprime (their GCD is a unit), then p must divide b. This property is deeply connected to the definition of prime elements.
• Schreier Domains: Due to the property that every nonzero element is primal, GCD domains are also known as Schreier domains. This classification further highlights their unique algebraic characteristics that ensure the equivalence of irreducible and prime elements.

These structural properties ensure that if an element cannot be factored into non-unit elements (irreducible), it must inherently satisfy the divisibility condition characteristic of prime elements.

Why This Matters

The equivalence of irreducible and prime elements in GCD domains is fundamental because it leads to important consequences regarding unique factorization. While not all GCD domains are unique factorization domains (UFDs) in the strictest sense (where elements factor uniquely into prime elements up to units and order), this property brings them very close. It ensures that the "building blocks" of elements (irreducible ones) behave in the same predictable way as prime numbers in arithmetic.

In summary, the rigorous structure of a GCD domain guarantees that its irreducible elements exhibit the strong divisibility properties characteristic of prime elements, making them one and the same in such domains.