Valuation and Pseudovaluation Subrings of an Integral Domain

A note on pairs of rings with same prime ideals

“…Let R ⊂ S be an extension of integral domains. Then R is a said to be a pseudo-valuation subring (PV, for short) in S, see [4], if for each x ∈ S \ R and non-unit a of R, we have x −1 a ∈ R. Note that PV in a domain is a generalization of PVD. We now continue to investigate the properties of ring extensions having the same set of prime ideals initiated in [1].…”

Maximal non valuation domains in an integral domain

“…Recall from [4], Remark 1.1 (3) that if R ⊂ S is an extension of integral domains and if R is a VD in S, then qf(R) = qf(S). Clearly, this may not be true if R is not a VD in S. However, if R is a maximal non VD in S, where R is not a field, then qf(R) = qf(S) as we have the next proposition which is a generalization of [7], Lemma 1 (ii).…”

“…DOI: 10.21136/CMJ.2020.0098-19 of terms of the longest maximal chains in Supp(S/R). Our work is motivated by [4] and [7]. Let R ⊂ S be a ring extension of integral domains.…”

“…Let R ⊂ S be a ring extension of integral domains. Then R is said to be a valuation subring of S (R is a VD in S for short), see [4], if whenever x ∈ S, we have x ∈ R or x −1 ∈ R. Note that if S = qf(R), then R is a valuation domain. Thus, the concept of valuation subrings of a domain is the generalization of valuation domains.…”

Comment on "Two notes on imbedded prime divisors"

“…The author was supported by the SRF grant from UGC India, Sr. No. 2061440976 2. The author was supported by the MATRICS grant from DST-SERB India, No.…”