Atul Gaur

On λ-extensions of commutative rings

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 this paper, a characterization of integrally closed λ-extension of rings is given. For example, we show that if R is a local ring, then R ⊆ T is an integrally closed λ-extension of rings if and only if there exists 𝔮 ∈Spec(R) such that T = R𝔮, 𝔮 = T𝔮 and R/𝔮 is a valuation domain. Let R be a subring of T such that R is invariant under action by G, where G is a subgroup of the automorphism group of T. If R ⊆ T is a λ-extension of rings, then RG ⊆ TG is a λ-extension of rings under some conditions.

Δ-Extension of Rings and Invariance Properties of Ring Extension under Group Action

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. I...

WHEN THE MAXIMAL GRAPH IS PLANAR, OUTERPLANAR, AND RING GRAPH

Let R be a commutative ring with nonzero identity. Let Γ(R) denote the maximal graph associated to R, that is, Γ(R) is a graph with vertices as non-units of R, where two distinct vertices a and b a...

Line Signed Graph of a Signed Total Graph

Abstract A signed total graph is an ordered pair T Σ ( Γ ( R ) ) : = ( T ( Γ ( R ) ) , σ ) , where T ( Γ ( R ) ) is the total graph of a commutative ring R, called the underlying graph of T Σ ( Γ ( R ) ) and T Σ ( Γ ( R ) ) is associated with a signing of its edges (a, b) by the function σ such that σ ( a , b ) is ‘+’ if a ∈ Z ( R ) or b ∈ Z ( R ) and ‘−’ otherwise. The aim of this paper is to gain a deeper insight into the notion of signed total graph by characterizing the rings for which line signed graph L ( T Σ ( Γ ( R ) ) ) of signed total graph are C-consistent, T Σ ( Γ ( R ) ) -consistent and sign-compatible.

𝒞-Consistency in signed total graphs of commutative rings

Motivated by the earlier study on the notion of signed total graph of a commutative ring, in this paper, we characterize all the commutative rings with unity for which signed total graph is 𝒞-consi...