How changing 𝐷[[𝑥]] changes its quotient field

Abstract

Let D [ [ x ] ] D[[x]] be the ring of formal power series over the commutative integral domain D D . It is shown that changing D [ [ x ] ] D[[x]] to D [ [ x / a ] ] D[[x/a]] changes (i.e. increases) the quotient field by an infinite transcendence degree over the original field whenever ∩ i = 1 ∞ a i D = 0 \cap _{i = 1}^\infty {a^i}D = 0 . From this it follows that if D 1 {D_1} and D 2 {D_2} are two distinct rings between the integers and the rational numbers, with D 1 {D_1} contained in D 2 {D_2} , then the change in the ring of coefficients from D 1 [ [ x ] ] {D_1}[[x]] to D 2 [ [ x ] ] D_{2}[[x]] again yields a change in the quotient fields by an infinite transcendence degree. More generally, it is shown that D D is completely integrally closed iff any increase in the ring of coefficients yields an increase in the quotient field of D [ [ x ] ] D[[x]] . Moreover, D D is a one-dimensional Prüfer domain iff any change in the ring of coefficients from one overring of D D to another overring of D D yields a change in the quotient field of the respective power series rings. Finally it is shown that many of the domain properties of interest are really properties of their divisibility groups, and some examples are constructed by first constructing the required divisibility groups.