Coherence and weak factoriality in a certain pullback (Q6561423)

All rings in this paper are commutative and unitary. Let \(D\) be an integral domain with quotient field \(K\), \(n\geq 2\) an integer, \(K[X]\) the univariate polynomial ring over \(K\) and \(\theta\) the image of \(X\) in the factor ring \(K[X]/(X^n)\). As \(D\) is a subring of \(K[\theta]\), consider the pullback ring \(R_n:=D+ \theta K[\theta]=\{ f\in K[\theta];\ f(\theta)\in D \}\). Most of the results in this paper follow the pattern ``\(R_n\) is q iff \(D\) is q'', with q being a ring property from a list including: coherent, regular coherent, locally coherent, quasi-coherent, w-coherent, v-coherent, regular weakly factorial, strong Mori, ring with Noetherian spectrum.