Evrim Akalan

Goldie Extending Modules

In this article, we define a module M to be 𝒢-extending if and only if for each X ≤ M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We consider the decomposition theory for 𝒢-extending modules and give a characterization of the Abelian groups which are 𝒢-extending. In contrast to the charac-terization of extending Abelian groups, we obtain that all finitely generated Abelian groups are 𝒢-extending. We prove that a minimal cogenerator for 𝒢od-R is 𝒢-extending, but not, in general, extending. It is also shown that if M is (𝒢-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.

Rings with Enough Invertible Ideals and Their Divisor Class Groups

We investigate Noetherian maximal orders with enough invertible ideals and their two different divisor class groups. We show that in a Noetherian maximal order R with enough invertible ideals, every height 1 prime ideal P is maximal reflexive and , where P ranges over all height 1 prime ideals of R, and S is a simple Noetherian ring. We show that one of the class groups of R measures, to some extent, the lack of unique factorisation in the ring. We also investigate relations between the class groups of R and the divisor class group of the center of R. Examples are provided to illustrate our results.

On Rings Whose Reflexive Ideals are Principal

We call a prime Noetherian maximal order R a pseudo-principal ring if every reflexive ideal of R is principal. This class of rings is a broad class properly containing both prime Noetherian pri-(pli) rings and Noetherian unique factorization rings (UFRs). We show that the class of pseudo-principal rings is closed under formation of n × n full matrix rings. Moreover, we prove that if R is a pseudo-principal ring, then the polynomial ring R[x] is also a pseudo-principal ring. We provide examples to illustrate our results.

Multiplicative Ideal Theory in Noncommutative Rings

The aim of this paper is to survey noncommutative rings from the viewpoint of multiplicative ideal theory. The main classes of rings considered are maximal orders, Krull orders (rings), unique factorization rings, generalized Dedekind prime rings, and hereditary Noetherian prime rings . We report on the description of reflexive ideals in Ore extensions and Rees rings. Further we give necessary and sufficient conditions (or sufficient conditions) for well-known classes of rings to be maximal orders, and we propose a polynomial-type generalization of hereditary Noetherian prime rings.

Goldie Extending Rings

In this article, we investigate the 𝒢-extending condition under various ring extensions. We show that if R R is 𝒢-extending and S is a right essential overring, then S R and S S are 𝒢-extending. For split-null extensions, we show that if M ⊴ R and M is left faithful, then R R is (𝒢-) extending if and only if S S is (𝒢-) extending, where S = S(R, M). This result appears to be new for the extending case. We conclude with results on Dorroh extensions.

Ore extensions over PI G-Dedekind prime rings

In this article, Ore extensions of the class of G-Dedekind prime rings satisfying a polynomial identity (PI) are studied. We prove that, if R is a PI G-Dedekind prime ring with an automorphism α of finite order, then R′ =R[x; α] is also a PI G-Dedekind prime ring.

Projective ideals of skew polynomial rings over HNP rings

ABSTRACT Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S = R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.

NEW CHARACTERIZATIONS OF GENERALIZED DEDEKIND PRIME RINGS

In this paper, we give new characterizations of generalized Dedekind prime rings in the class of prime Noetherian rings satisfying a polynomial identity. We provide examples to illustrate our results.

Corrigendum To: Goldie Extending Modules

This corrigendum is written to correct the proof of Theorem 5.3 of Akalan et al. [1].

Generalized hereditary Noetherian prime rings

The polynomial rings over hereditary Noetherian prime rings have global dimension 2 and any two-sided ideal which is either left v-ideal or right v-ideal is left and right projective. By using the latter property, we define the concept of generalized hereditary Noetherian prime rings (G-HNP rings). We study the structure of projective ideals in G-HNP rings and some overrings of G-HNP rings. As it is shown in the examples, the class of G-HNP rings ranges from rings with global dimension 2 to rings with infinite global dimension and Noetherian prime rings with global dimension 2 are not necessarily G-HNP rings.