An instance I of Ring Grooming consists of m sets A1, A2, . . . , Am from the universe {0, 1, . . . , n − 1} and an integer g ≥ 2. The unrestricted variant of Ring Grooming, referred to as Unrestricted Ring Grooming, seeks a partition {P1, P2, . . . , Pk} of {1, 2, . . . , m} such that |Pi | ≤ g for each 1 ≤ i ≤ k and ∑k i=1 | ⋃ r∈Pi Ar | is minimized. The restricted variant of Ring Grooming, r...

Let S be a cancellative monoid with quotient group of torsion-free rank a. We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a. Assume that R is a commutative unitary ring and G is an abelian group. The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the gr...

Let G be a finite simple graph on the vertex set $$V(G) = \{x_{1}, \ldots , x_{n}\}$$ and match(G), min-match(G) ind-match(G) matching number, minimum number induced of G, respectively. $$K[V(G)] K[x_{1}, x_{n}]$$ denote polynomial ring over field K $$I(G) \subset K[V(G)]$$ edge ideal G. The relationship between these graph-theoretic invariants ring-theoretic quotient K[V(G)]/I(G) has been stud...

‎Let $R$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$U$ the Utumi quotient ring of $R$‎, ‎$C$ $(=Z(U))$ the extended centroid‎ ‎of $R$‎. ‎Let $0neq ain R$ and $f(x_1,ldots,x_n)$ a multilinear‎ ‎polynomial over $C$ which is noncentral valued on $R$‎. ‎Suppose‎ ‎that $G$ and $H$ are two nonzero generalized derivations of $R$‎ ‎such that $a(H(f(x))f(x)-f(x)G(f(x)))in ...

Given a Lie superalgebra g, we introduce several variants of the representation ring, built as subrings and quotients of the ring RZ2(g) of virtual g-supermodules, up to (even) isomorphisms. In particular, we consider the ideal R+(g) of virtual g-supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring SR(g) on which the parity rever...

Let $R$ be a 2-torsion free ring and $U$ be a square closed Lie ideal of $R$. Suppose that $alpha, beta$ are automorphisms of $R$. An additive mapping $delta: R longrightarrow R$ is said to be a Jordan left $(alpha,beta)$-derivation of $R$ if $delta(x^2)=alpha(x)delta(x)+beta(x)delta(x)$ holds for all $xin R$. In this paper it is established that if $R$ admits an additive mapping $G : Rlongrigh...

The strong ring R is a commutative ring generated by finite abstract simplicial complexes. To every G ∈ R belongs a Hodge Laplacian H = H(G) = D = (d + d∗)2 determining the cohomology and a unimodular connection operator L = L(G). The sum of the matrix entries of g = L−1 is the Euler characteristic χ(G). For any A,B ∈ R the spectra of H satisfy σ(H(A × B)) = σ(H(A)) + σ(H(B)) and the spectra of...