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Let D be a unique factorization domain and S an infinite subset of D. If f (X) is an element in the ring of integer-valued polynomials over S with respect to D (denoted Int(S, D)), then we characterize the irreducible elements of Int(S, D) in terms of the fixed-divisor of f (X). The characterization allows us to show that every nonzero rational number n/m is the leading coefficient of infinitely many irreducible polynomials in the ring Int(Z) = Int(Z, Z). Further use of the characterization leads to an analysis of the particular factorization properties of such integer-valued polynomial rings. In the case where D = Z, we are able to show that every rational number greater than 1 serves as the elasticity of some polynomial in Int(S, Z) (i.e., Int(S, Z) is fully elastic).

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