The following result was proved in [5,Remark 2.2].
Theorem 0.1. If R T are Noetherian rings such that there does not
exist any integrally dependent adjacent Noetherian rings between them, then
for each ¯c/¯b 2 T/Z ...
Let R be a commutative ring with identity. The ring R × R can be
viewed as an extension of R via the diagonal map : R →֒ R×R, given
by (r) = (r, r) for all r ∈ R. It is shown that, for any a, b ∈ R, the
extension ...
The notion of maximal non valuative domain is introduced and characterized.
An integral domain R is called a maximal non valuative domain
if R is not a valuative domain but every proper overring of R is
a valuative ...
Let R be a commutative ring with unity. The notion of maximal non valuation
domain in an integral domain is introduced and characterized. A proper subring R of an
integral domain S is called a maximal non valuation domain ...
Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained ...
This paper is a sequel. The earlier paper introduced, for any (unital) extension of (commutative
unital) rings R T, an invariant L(T=R) defined as the supremum of the lengths of chains of intermediate
fields in the ...
In this note, we show that a part of Ratliff (Proc Am Math Soc 101(3):395–402, 1987, Remark 2.2) is not correct. Some conditions are given under which the same holds.
Let H denotes the set of all commutative rings R in which the set of all nilpotent elements, denoted by Nil(R), is a prime ideal of R and is comparable to every ideal of R. Let R∈H be a ring and T(R) be its total quotient ...
The notion of maximal non-ϕ-pseudo-valuation ring is introduced which generalizes the concept of maximal non-pseudo-valuation domain. The equivalence of maximal non-ϕ-PVR and maximal non-local ring is established under ...
Kumar, Rahul(The Korean Mathematical Society., 2023-01)
Let H0 be the set of rings R such that Nil(R)=Z(R) is a divided prime ideal of R. The concept of maximal non ϕ-chained subrings is a generalization of maximal non valuation subrings from domains to rings in H0. This ...
Let H0 denote the set of all rings R such that Nil(R) is a divided prime ideal with Nil(R)=Z(R). We study the concept of maximal non-λ-rings in class H0 and generalize the results of maximal non-λ-domains.
Let R be a commutative ring with identity. In A. Azarang, O. A. S. Karamzadeh, and A. Namazi, [Ukr. Math. J., 65, No. 7, 981–994 (2013) (Proposition 3.1)], it was proved that if R is an integral domain and S is a maximal ...
Let R be a commutative ring with unity. The notion of maximal non -subrings
is introduced and studied. A ring R is called a maximal non -subring of a ring T if R T
is not a -extension, and for any ring S such that ...
Kumar, Rahul(The Belgian Mathematical Society, 2020)
Let R be an integral domain. Then R is said to be a λ-domain if the set of all overrings of R is linearly ordered by inclusion. If R1+R2 is an overring of R for each pair of overrings R1,R2 of R, then R is said to be a ...
Let R be a commutative ring with unity. Let H denotes the set of all rings R such that Nil(R) is a divided prime ideal. The notion of maximal non-Prüfer ring and maximal non-ϕ-Prüfer ring is introduced which generalize the ...
Chiral nanophotonic platforms provide a means of creating near fields with
both enhanced asymmetric properties and intensities. They can be exploited
for optical measurements that allow enantiomeric discrimination at ...
Let R be a commutative ring with unity. The notion of λ-rings, Φ-λ-rings, and Φ-Δ-rings is introduced which generalize the concept of λ-domains and Δ-domains. A ring R is said to be a λ-ring if the set of all overrings of ...
We study the ring extensions R⊆T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exists properly containing R. Using idealization theory, the ...
Let R,T be commutative rings with identity such that R⊆T. A ring extension R⊆T is called a Δ-extension of rings if R1+R2 is a subring of T for each pair of subrings R1,R2 of T containing R. In this paper, a characterization ...