On sums and products of primitive elements

摘要

Let R be a (commutative integral) domain, with K its quotient field and R its integral closure (in K). Let be the set of elements uK such that u is primitive over R; i.e., such that u is the root of a polynomial over R having a unit coecient. Then, is a ring (necessarily K) double left right arrow is closed under products double left right arrow R is a Prufer domain. In general, is closed under powers. For u,v, necessary and sucient conditions are given for u+v (resp., uv) to belong to . Also, is used to characterize when R is a quasi-local integrally closed domain and when R is a pseudo-valuation domain. If R is quasi-local, each element of K is expressible as the sum of two (possibly equal) elements of . The set of primitive elements is determined for lying-over pairs and for extensions of domains with the same sets of prime ideals. In this study of the construction, R and K are replaced, whenever possible, by an arbitrary commutative ring and its total quotient ring or, more generally, by any inclusion of commutative rings.