The Fundamental Theorem of Arithmetic shows that an integer greater than 1 can be uniquely factorized into primes; a unique factorization domain is essentially a ring in which the fundamental theorem holds. Specifically, an integral domain R is a UFD if

1. (i) every non-zero element, which is not a unit, can be written as a product of irreducible elements;

2. (ii) if p₁p₂…p_m = q₁q₂…q_n, where the p_i and q_j are irreducible, then m = n and (with possible reordering) for each i we have p_i = u_iq_i for some unit u_i.

In a UFD, an element is prime if and only if it is irreducible. If R is a UFD, then R[x] is also a UFD. Euclidean domains are UFDs.

Note in $R=\left\{a+b\sqrt{-3}\right\}$ that $4=2\times 2=(1+\sqrt{-3})(1-\sqrt{-3})$ are two essentially different factorizations of 4, and so R is not a UFD. By contrast, ℝ[x], the ring of real polynomials in a variable x, is a UFD. Note that 2x² − 6x + 4 = (2x − 2)(x − 2) =(x − 1)(2x − 4), but these are essentially the same factorization as (2x − 2) = 2(x − 1) and (x − 2) = 2⁻¹(2x − 4), and 2 is a unit in ℝ[x].