‎Let $D$ be an integral domain with quotient field $K$‎, ‎$E$ be a $K$-vector space‎, ‎$R = D propto E$ be the trivial extension of $D$ by $E$‎, ‎and $w$ be the so-called $w$-operation‎. ‎In this paper‎, ‎we show that‎ ‎$R$ is a $w$-FF ring if and only if $D$ is a $w$-FF domain; and‎ ‎in this case‎, ‎each $w$-flat $w$-ideal of $R$ is $w$-invertible.

Abstract We introduce the notion of a product fractal ideal ring using permutations finite sets and multiplication operation in ring. This generalizes concept an obtain corresponding quotient structure that partitions under certain conditions. prove isomorphism theorems illustrate involved with examples. These extend classical rings, providing broader viewpoint.

The concept of convex ordered hyperrings associated with a strongly regular relation was investigated in this study. In paper, we first studied hyperatom elements and then characterizations quotient rings. Is there θ on hyperring R for which R/θ is ring? This leads to an ring obtained from hyperring.

If V is an affine algebraic variety and G ⊂ AutV a finite group of automorphism of V , the quotient variety is an affine algebraic variety V/G with a quotient morphism V → X = V/G. A point of X is an orbit of G on V , and the coordinate ring k[X] is the ring of invariants k[V ] of the induced action of G on k[V ]. This chapter studies the simplest case of this construction, when V = C and G = Z...

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...

We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description...

We show that the symmetric product of a flat affine scheme over a commutative ring can be embedded into the quotient by the general linear group of the scheme of commuting matrices. We also prove that the symmetric product of the affine space is isomorphic to the above quotient when the base ring is a characteristic zero field. Over an infinite field of arbitrary characteristic the quotient of ...

Let R be the universal deformation ring of a residual representation of a local Galois group. Kisin showed that many loci in MaxSpec(R[1/p]) of interest are Zariski closed, and gave a way to study the generic fiber of the corresponding quotient of R. However, his method gives little information about the quotient ring before inverting p. We give a method for studying this quotient in certain ca...

For a connected reductive group G and a finite–dimensional G–module V , we study the invariant Hilbert scheme that parameterizes closed G–stable subschemes of V affording a fixed, multiplicity–finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T–stable subscheme admitting a good quotient. ...