There exists an analogy between the structure of ideals of a cone in a right-ordered group and the structure of ideals of a Dubrovin valuation ring in a simple artinian ring. The structure of ideals of rank one cones and rank one Dubrovin valuation rings can be described completely. One sided versions of this problem are considered in 5] where the ideal theory of right cones is developed and a ...

We show that the quasiequational theory of a relatively congruence modular quasivariety of left R-modules is determined by a two-sided ideal in R together with a filter of left ideals. The two-sided ideal encodes the identities that hold in the quasivariety, while the filter of left ideals encodes the quasiidentities. The filter of left ideals defines a generalized notion of torsion. It follows...

Let I be a homogeneous Artinian ideal in a polynomial ring R = k[x1, . . . , xn] over a field k of characteristic 0. We study an equivalent condition for the generic initial ideal gin(I) with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that Moreno-Socias conjecture implies Fröberg conjecture. And for the case codimI ≤ 3, we show that R/I has t...

Let R be an associative ring. A map σ : R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy) = σ(x)σ(y) for all elements a,b ∈ R. A ring R is said to be rigid if it has only the trivial ring endomorphisms, that is, identity idR and zero 0R . Rigid left Artinian rings were described by Maxson [9] and McLean [11]. Friger [4, 6] has constructed an example of a noncommutative rigid r...

Let [Formula: see text] be a commutative ring and nonzero text]-module. We introduce the class of pseudo-strongly (PS)-hollow submodules text]. Inspired by theory modules with secondary representations, we investigate which can written as finite sums PS-hollow submodules. In particular, provide existence uniqueness theorems for minimal strongly representations over Artinian rings.

An R-module M is called strongly noncosingular if it has no nonzero Rad-small (cosingular) homomorphic image in the sense of Harada. It is proven that (1) an R-module M is strongly noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring R is Artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

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...

A module $$M$$ is said to be distributive (resp., uniserial) if the submodule lattice of a chain) Any uniserial but ring integers non-uniserial as $$\mathbb{Z}$$ -module. Direct sums (resp. modules are called semidistributive serial) modules. If $$A$$ with automorphism $$\varphi$$ , then we denote by $$A((x,\varphi))$$ skew Laurent series coefficient in which addition naturally defined and mult...

let $r$ be a commutative ring with identity. an ideal $i$ of a ring $r$is called an annihilating ideal if there exists $rin rsetminus {0}$ such that $ir=(0)$ and an ideal $i$ of$r$ is called an essential ideal if $i$ has non-zero intersectionwith every other non-zero ideal of $r$. thesum-annihilating essential ideal graph of $r$, denoted by $mathcal{ae}_r$, isa graph whose vertex set is the set...

Let R = ⊕ n≥0 Rn be a homogeneous Noetherian ring with local base ring (R0,m0) and let M be a finitely generated graded R-module. Let Hi R+ (M) be the i-th local cohomology module of M with respect to R+ := ⊕ n>0 Rn. If dimR0 ≤ 1, the R-modules Γm0R(H R+ (M)), (0 :Hi R+ (M) m0) and Hi R+ (M)/m0H i R+ (M) are Artinian for all i ∈ N0. As a consequence, much can be said on the asymptotic behaviour...