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Let D be an integral domain with quotient field K and let X be an indeterminate over D. Also, let T:={Tλ|λ∈Λ} be a defining family of quotient rings of D and suppose that * is a finite type star operation on D induced by T. We show that D is a P*MD (respectively, PvMD) if and only if (cD(fg))*=(cD(f)cD(g))* (respectively, (cD(fg))w=(cD(f)cD(g))w) for all 0≠f,g∈K[X]. A more general version of this...
For the domain R arising from the construction T, M, D, we relate the star class groups of R to those of T and D. More precisely, let T be an integral domain, M a nonzero maximal ideal of T, D a proper subring of k:=T/M, φ:T→k the natural projection, and let R=φ−1(D). For each star operation * on R, we define the star operation *φ on D, i.e., the “projection” of * under φ, and the star operation (*)T...
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