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 ...
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 ...
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. 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 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 ...
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 -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 ...
Let R be a commutative ring with unity. The notion of maximal non chained subrings of a ring and maximal non ϕ-chained subrings of a ring is introduced which generalizes the concept of maximal non valuation subrings of 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 ...
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 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 and S be a (unital) subring of R such that R is integral over S and S⊆R has FCP. Let M be an R-module. For any submodule N of M, it is shown that R(+)N⊆R(+)M has FCP if and only if ...
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 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.
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 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 ...
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 ...
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 ...
Let R,T be commutative rings with identity such that R⊆T. We recall that R⊆T is called a λ-extension of rings if the set of all subrings of T containing R (the “intermediate rings”) is linearly ordered under inclusion. In ...