1,721,004 research outputs found
Essential domains and two conjectures in dimension theory
No full textThis note investigates two long-standing conjectures on the Krull
dimension of integer-valued polynomial rings and of polynomial rings, respectively, in the context of (locally) essential domains
ON THE KRULL AND VALUATIVE DIMENSION OF D+XDS[X] DOMAINS
No full textAbstractIn this paper, we deal with the integral domain D(S,r):=D+(X1,X2,…,Xr)DS[X1, X2,…,Xr], where D is an integral domain and S is a multiplicative set of D. The purpose is to pursue the study, initiated by Costa-Mott-Zafrullah in 1978, concerning the prime ideal structure of such domains. We characterize when D(S,r) is a strong S-domain, a stably strong S-domain, a catenarian domain and a universally catenarian domain. As a consequence, we obtain a new class of non-Noetherian universally catenarian domains. Moreover, we give an explicit formula for the Krull dimension of D(S,r) (depending on S and on the Krull dimensions of D and DS[X1,X2,…,Xr]) and we compute its valuative dimension
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