Right Gaussian rings and skew power series rings

AbstractWe introduce a class of rings we call right Gaussian rings, defined by the property that for any two polynomials f, g over the ring R, the right ideal of R generated by the coefficients of the product fg coincides with the product of the right ideals generated by the coefficients of f and of g, respectively. Prüfer domains are precisely commutative domains belonging to this new class of rings. In this paper we study the connections between right Gaussian rings and the classes of Armendariz rings and rings whose right ideals form a distributive lattice. We characterize skew power series rings (ordinary as well as generalized) that are right Gaussian, extending to the noncommutative case a well-known result by Anderson and Camillo. We also study quotient rings of right Gaussian rings

The minimal prime spectrum of rings with annihilator conditions

AbstractIn this paper, we study rings with the annihilator condition (a.c.) and rings whose space of minimal prime ideals, Min(R), is compact. We begin by extending the definition of (a.c.) to noncommutative rings. We then show that several extensions over semiprime rings have (a.c.). Moreover, we investigate the annihilator condition under the formation of matrix rings and classical quotient rings. Finally, we prove that if R is a reduced ring then: the classical right quotient ring Q(R) is strongly regular if and only if R has a Property (A) and Min(R) is compact, if and only if R has (a.c.) and Min(R) is compact. This extends several results about commutative rings with (a.c.) to the noncommutative setting

Hopf Galois Extensions and Non-commutative Rings

In this document we review the notion of Hopf Galois extensions, addressing remarkable examples, some properties and recent advances in such theory. Similarly, we present some families of non-commutative rings and algebras which arise in several applications and contexts. With particular examples, and using recent results on Hopf Galois theory, we study some interactions between such extensions and the mentioned families.Resumen: En este documento abordaremos la noción de extensión Hopf Galois, revisando los ejemplos más destacados, algunas propiedades y los avances recientes en dicha teoría. De igual manera, presentamos algunas familias de anillos y álgebras no-conmutativas que aparecen en diversas aplicaciones y escenarios. Con ejemplos particulares, y usando trabajos recientes en teoría Hopf Galois, estudiamos algunas interacciones entre dichas extensiones y algunas de las familias mencionadas.Maestrí