  The article gives a ring theoretic perspective on cluster algebras.
Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster
algebra are irreducible elements. Furthermore, they provide two necessary
conditions for a cluster algebra to be a unique factorization domain, namely
the irreducibility and the coprimality of the initial exchange polynomials.
  We present a sufficient condition for a cluster algebra to be a unique
factorization domain in terms of primary decompositions of certain ideals
generated by initial cluster variables and initial exchange polynomials. As an
application, the criterion enables us to decide which coefficient-free cluster
algebras of Dynkin type A, D or E are unique factorization domains. Moreover,
it yields a normal form for irreducible elements in cluster algebras that
satisfy the condition. Proof techniques include methods from commutative
algebra.
  In addition, we state a conjecture about the range of application of the
criterion.
