Rings of continuous functions in which every finitely generated ideal is principal

Rings of Continuous Functions in Which Every Finitely Generated Ideal is Principal

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prüfer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical...

Finitely Generated Annihilating-Ideal Graph of Commutative Rings

Let $R$ be a commutative ring and $mathbb{A}(R)$ be the set of all ideals with non-zero annihilators. Assume that $mathbb{A}^*(R)=mathbb{A}(R)diagdown {0}$ and $mathbb{F}(R)$ denote the set of all finitely generated ideals of $R$. In this paper, we introduce and investigate the {it finitely generated subgraph} of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_F(R)$. It is the (undi...

Rings for which every simple module is almost injective

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

Computational ideal theory in finitely generated extension rings

One of the most general extensions of Buchberger's theory of Grobner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Gr obner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of nitely generated extension rings and present subclasses of such structur...