The complete quotient ring of images of semilocal Prüfer domains

John Chuchel, Norman Eggert
1977 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Introduction. It is well known that the complete quotient ring of a Noetherian ring coincides with its classical quotient ring, as shown in Akiba [1] . But in general, the structure of the complete quotient ring of a given ring is largely unknown. This paper investigates the structure of the complete quotient ring of certain Prùfer rings. Boisen and Larsen [2] considered conditions under which a Prùfer ring is a homomorphic image of a Prùfer domain and the properties inherited from the domain.
more » ... e restrict our investigation primarily to homomorphic images of semilocal Prùfer domains. We characterize the complete quotient ring of a semilocal Prùfer domain in terms of complete quotient rings of local rings and a completion of a topological ring. Further, if the kernel of the homomorphism has an irredundant primary decomposition, we characterize the elements of the complete quotient ring. Throughout the paper, all rings are commutative and have identity 1. If 5 is a multiplicatively closed set in a ring R, we let Rs be a ring of quotients of R. For S the set of regular elements of R, Rs is Q C \(R), the classical quotient ring of R. If S is the complement of a prime ideal P of R, Rs is also written as R P} the localization of R to P. Among the conditions which are equivalent to R being a Prùfer domain, one which we will find particularly useful is:
doi:10.4153/cjm-1977-092-x fatcat:2eo5gizyj5fvlkgnglau6kvjqe