On the irreducibility of pure difference binomials (Q1176617)

It is well known that the class of monomial rings \(R[S]\), \(R\) a ring and \(S\) a commutative semigroup, coincides with the class \(R[X]/I\), where \(X\) is a set of indeterminates and \(I\) is an ideal generated by difference binomials \(m_ i-m_ j\), where \(m_ i\), \(m_ j\) are monomials. The author is interested in the ring \(D[X_ 1,\dots,X_ m,Y_ 1,\dots,Y_ n]/(f)\), where \(f=X_ 1^{a_ 1}\dots X_ m^{a_ m}-Y_ 1^{b_ 1}\dots Y_ n^{b_ n}\); here \(D\) is a domain, and \(a_ 1,\dots,a_ m\), \(b_ 1,\dots,b_ n\) are positive integers. Let \(s:=\text{gcd}(a_ 1,\dots,a_ m,b_ 1,\dots,b_ n)\). The author proves (theorem 5) that \(f\) is reducible in \(D[X_ 1,\dots,Y_ n]\) if \(s>1\), and that \(f\) is a prime ideal in \(D[X_ 1,\dots,Y_ n]\) if \(s=1\). Moreover (theorem 6), if \(s=1\), he shows that \(D[X_ 1,\dots,Y_ n]/(f)\) is a monoid domain.