A note on denominator ideals of linear fractional transforms of an anti-integral element over an integral domain

Let α be an anti-integral element of degree t over an integral domain R and φα(X) the minimal polynomial of α over the quotient field of R. Let β be a linear fractional transform of α, that is, β=cα-d/aα-b(a, b, c, d∈R, ad-bc∈R*)where R* is the group of units of R. First we describe I[β], the denominator ideal of β, in terms of I[α] and φα(a, b) where φα(X, Y)=Xtφα(Y/X). Next we introduce the ideal ˜{I}[α] concerning integral property of α and α-1. Then we describe ˜{I}[β] by using I[α], φα(a, b) and φα(c, d).