Let S be a finitely generated standard multigraded algebra over an Artinian local ring A; M a finitely generated multigraded S-module. This paper answers to the question when mixed multiplicities of M are positive and characterizes them in terms of lengths of A-modules. As an application, we get interesting results on mixed multiplicities of ideals, and recover some early results in [Te] and [TV].

The aim of this paper is to investigate strong notion strongly ⨁-supplemented modules in module theory, namely ⨁-locally artinian supplemented modules. We call a M if it locally and its supplement submodules are direct summand. In study, we provide the basic properties particular, show that every summand supplemented. Moreover, prove ring R semiperfect with radical only projective R-module

We consider the following condition (*) on an associative ring R : (*). There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R2, and f(xy) = (xy)n(x,y) for some positive integer n(x,y) > 1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for nonArtinian rings. The results generalize...

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answe...

Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period 2 . It is proved however, that the...

let $r$ be a commutative ring with identity and let $m$ be an $r$-module. a proper submodule $p$ of $m$ is called strongly prime submodule if $(p + rx : m)ysubseteq p$ for $x, yin m$, implies that $xin p$ or $yin p$. in this paper, we study more properties of strongly prime submodules. it is shown that a finitely generated $r$-module $m$ is artinian if and only if $m$ is noetherian and every st...

In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime i...

Chow rings of matroids were instrumental in the resolution Heron–Rota–Welsh Conjecture by Adiprasito, Huh, and Katz Top-Heavy Braden, Matherne, Proudfoot, Wang. The ring a matroid is commutative, graded, Artinian, Gorenstein algebra with linear quadratic relations defined matroid. Dotsenko conjectured that any Koszul. purpose this paper to prove Dotsenko’s conjecture. We also show augmented As ...

We show that if R is an exchange ring with primitive factors artinian then K1(R) U(R)/V(R), where U(R) is the group of units of R and V(R) is the subgroup generated by {(1+ab)(1+ba)−1 | a,b ∈ R with 1+ab ∈ U(R)}. As a corollary, K1(R) is the abelianized group of units of R if 1/2∈ R. 2000 Mathematics Subject Classification. 16E50, 19B10. Very recently, Ara et al. [2] showed that the natural hom...

Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected grap...