arXiv subjects
Publications and source records attributed to Mohsen Aghajani.
In this paper, we consider the N-pure notion. An ideal $I$ of a ring $R$ is said to be N-pure, if for every $a\in I$ there exists $b\in I$ such that $a(1-b)\in N(R)$, where N(R) is nil radical of $R$. We provide new characterizations for N-pure ideals. In addition, N-pure ideals of an arbitrary ring are identified. Also, some other properties of N-pure ideals are studied. finally, we prove some results about the endomorphism ring of pure and N-pure ideals.
In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.
In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. A new class of rings is introduced and studied, we call them purified rings. Specially, reduced purified rings are characterized. New characterizations for pure ideals of reduced Gelfand rings and mp-rings are provided. It is also proved that if the topology of a scheme is Hausdorff, then the affine opens of that scheme is stable under taking finite unions. In particular, every compact scheme is an affine scheme.
In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between the idempotents of a ring and the clopens of its pure spectrum. It is also proved that a given ring is a Gelfand ring iff its maximal spectrum equipped with the induced Zariski topology is homeomorphic to its pure spectrum. Then as an application, it is deduced that a ring is zero dimensional iff its prime spectrum and pure spectrum are isomorphic. Dually, it is shown that a given ring is a reduced mp-ring iff its minimal spectrum equipped with the induced flat topology and its pure spectrum are the same. Finally, the new notion of semi-Noetherian ring is introduced and Cohen type theorem is proved.
In this paper, a combination of algebraic and topological methods are applied to obtain new and structural results on harmonic rings. Especially, it is shown that if a Gelfand ring $A$ modulo its Jacobson radical is a zero dimensional ring, then $A$ is a clean ring. It is also proved that, for a given Gelfand ring $A$, then the retraction map Spec$(A)\rightarrow{\rm Max}(A)$ is flat continuous if and only if $A$ modulo its Jacobson radical is a zero dimensional ring. Dually, it is proved that for a given mp-ring $A$, then the retraction map Spec$(A)\rightarrow{\rm Min}(A)$ is Zariski continuous if and only if ${\rm Min}(A)$ is Zariski compact. New criteria for zero dimensional rings, mp-rings and Gelfand rings are given. The new notion of lessened ring is introduced and studied which generalizes "reduced ring" notion. Specially, a technical result is obtained which states that the product of a family of rings is a lessened ring if and only if each factor is a lessened ring. As another result in this spirit, the structure of locally lessened mp-rings is also characterized. Finally, it is characterized that a given ring when is a finite product of fields, integral domains, local rings, and lessened quasi-prime rings.
In this paper, the new concept of quasi-prime ideal is introduced which at the same time generalizes the `prime ideal' and `primary ideal' notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which generalizes the Zariski topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is proved that if the Grothendieck t-functor is applied on the quasi-prime spectrum then the prime spectrum is deduced. It is also shown that there are the cases that the prime spectrum and quasi-prime spectrum do not behave similarly.
Let $R$ be a commutative Noetherian ring. In this paper, we study those finitely generated $R$-modules whose Cousin complexes provide Gorenstein injective resolutions. We call such a module a G-Gorenstein module. Characterizations of G-Gorenstein modules are given and a class of such modules is determined. It is shown that the class of G-Gorenstein modules strictly contains the class of Gorenstein modules. Also, we provide a Gorenstein injective resolution for a balanced big Cohen-Macaulay $R$-module. Finally, using the notion of a G-Gorenstein module, we obtain characterizations of Gorenstein and regular local rings.