9 papers
A counterexample to a question on the maximality of purely-primes
Abolfazl Tarizadeh
By using an advanced model of AI, it is proved that there exists a commutative ring having a purely-prime ideal which is not purely-maximal. This gives a counterexample to \cite[Co…
Ideal class group of an extension of rings and Picard group
Abolfazl Tarizadeh
For any extension of commutative rings , by using invertible ideals, we first define an Abelian group $\Cl(A,B)$, that we call the ideal class group of this extension…
Splitting in a complete local ring and decomposition its group of units
Abolfazl Tarizadeh
Let be a complete local ring (not necessarily Noetherian). As the first main result of this article, we prove that in the unequal characteristic case $\Char(R)\neq\Char(k…
Connected components of qcqs schemes and projective spaces
Abolfazl Tarizadeh
In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \\ As an application, the structure of…
Grading of homogeneous localization by the Grothendieck group
Abolfazl Tarizadeh
The main result of this article is a fantastic generalization of a classical result in graded ring theory. In fact, our result states that if is a multiplicative set of homogen…
On the Grothendieck ring and the relation of its group of units with the Picard group
Abolfazl Tarizadeh
As the first main result of this article, we prove that if and are idempotents of a commutative ring , then there is a canonical isomorphism of -modules: $$Ae\oplus…