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 be a Noetherian ring and I an ideal in R. Then there exists an integer k such that for all n 1 there exists a primary decomposition I n = q 1 \ \ q s such that for all i, p q i nk q i. Also, for each homogeneous ideal I in a polynomial ring over a eld there exists an integer k such that the Castelnuovo-Mumford regularity of I n is bounded above by kn. The regularity part follows from the ...

Following an idea of Kronecker, we describe a method for factoring prime ideal extensions in number rings. The method needs factorization of polynomials in many variables over finite fields, but it works for any prime and any number field extension. Introduction Let F c K be number fields, let ¿fp c <?* be their corresponding number rings, i.e., the integral closures of Z in F and K, respective...

let $f: arightarrow b$ be a ring homomorphism and let $j$ be an ideal of $b$. in this paper, we investigate the transfer of the property of coherence to the amalgamation $abowtie^{f}j$. we provide necessary and sufficient conditions for $abowtie^{f}j$ to be a coherent ring.

We show that minimal pairs exist in the quotient structure of R modulo the ideal of noncuppable degrees. In the study of mathematical structures it is very common to form quotient structures by identifying elements in some equivalence classes. By varying the equivalence relations, the corresponding quotient structures often reveal certain hidden features of the original structure. In this paper...