An integral domain $D$ is called a emph{locally GCD domain} if $D_{M}$ is aGCD domain for every maximal ideal $M$ of $D$. We study somering-theoretic properties of locally GCD domains. E.g., we show that $%D$ is a locally GCD domain if and only if $aDcap bD$ is locally principalfor all $0neq a,bin D$, and flat overrings of a locally GCD domain arelocally GCD. We also show that the t-class group...

Proposition 0.1. Let R be a commutative ring. TFAE: 1. Every ideal in R is finitely generated. 2. Every chain of ideals I1 ⊆ I2 ⊆ · · · is stable. That is, there exists n such that In = In+1 = · · · . 3. Every nonempty set of ideals has a maximal element wrt inclusion. Proof. 1. (1) =⇒ (2). I = ⋃ k Ik ideal in R. I is generated by x1, x2, . . . xm. There exists n such that x1, . . . , xm ∈ In. ...

In this paper, we study MV?algebra of continuous functions C(X) and maximal ideals C(X). Furthermore, Z?ideal Z??ideal are introduced proved that every in is a but the converse not true finitely generated basic Z??ideal. Also, investigate meet join two Z?ideals (Z??ideal) Complemented elements examined their properties have been studied. particular, relationship between ideal by them (Z??ideals...

In this paper, we define the notion of minimal prime ideals hoops and investigate some properties them. Then by using annihilators, study relation between annihilators. Also, introduce zero divisors elements prove that set all is a union hoop. Finally, notions maximal hoop, two new as p-ideal m-ideal. them every semi-simple hoop an m-ideal it.

We study non-holonomic overideals of a (left) differential ideal J ⊂ F [dx, dy ] in two variables where F is a differentially closed field of characteristic zero. The main result states that a principal ideal J = 〈P 〉 generated by an operator P with a separable symbol symb(P ) (being a homogeneous polynomial in two variables) has a finite number of maximal non-holonomic overideals. This stateme...

A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of  $R^n$ which is generated by columns of  a matrix $A=(a_{ij})$ with $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set.  ...

Let R be a Cohen-Macaulay local ring with maximal ideal m. In this paper we present a procedure for computing the Ratllif-Rush closure of a m−primary ideal I ⊂ R.

Let R be a Cohen-Macaulay local ring with maximal ideal m. In this paper we present a procedure for computing the coefficient ideals, in particular the Ratllif-Rush closure, of a m−primary ideal I ⊂ R.