New results about normal pairs of rings with zero-divisors

Ring Extensions with Finitely Many Non-Artinian Intermediate Rings

“…As R ⊂ R[a] is an integral extension and R[a] is zero dimensional, we infer that so too is R. It is also evident that R is Noetherian, and hence R is Artinian, which is a contradiction. We conclude using [25] ( eorem 1), see also the last comments in our introduction, that (R, S) is a normal pair. We will demonstrate that R ⊂ S is a minimal ring extension.…”

“…So, several characterizations of such pairs have been obtained. In [25][26][27], the authors have studied normal pairs of rings with zero divisors, so many results are generalized from the domain-theoretic case to arbitrary rings.…”

“…As in [29], if R⊆S is a ring extension, we say that (R, S) is an INC pair if R⊆T satisfies INC for each S-overring T of R. It was proved in [30] ( eorem) that R⊆S is a P-extension if and only if (R, S) is an INC pair. e authors in [25] ( eorem 1) ensure that (R, S) is a normal pair if and only if R ⊂ S is a P-extension and R is integrally closed in S. e next theorem characterizes ring extensions with only one non-Artinian intermediate ring.…”

“…en, according to Proposition 2, there exists an intermediate ring T such that R ⊂ T and T ⊂ S are minimal extensions, S is Artinian, and T is not Artinian. In view of [25] (2)⇒(1). If R ⊂ S is a minimal extension with S non-Artinian, then according to [1] ( eorem 2), R is not Artinian.…”