Alveri Sant'Ana - Academia.edu (original) (raw)
Uploads
Papers by Alveri Sant'Ana
Publicationes mathematicae
In this paper we introduce the notions of partial bimodule algebra and partial bico- module algeb... more In this paper we introduce the notions of partial bimodule algebra and partial bico- module algebra. We also deal with the existence of globalizations for these structures, generalizing related results appeared in [2, 4]. As an application we construct the partial (L;R)-smash product, extending the corresponding global notion appeared in [14] to the context of partial Hopf actions.
Journal of Pure and Applied Algebra, 2015
ABSTRACT In this paper we introduce the notion of partial action of a weak Hopf algebra on algebr... more ABSTRACT In this paper we introduce the notion of partial action of a weak Hopf algebra on algebras, unifying the notions of partial group action [11], partial Hopf action ([2],[3],[9]) and partial groupoid action [4]. We construct the fundamental tools to develop this new subject, namely, the partial smash product and the globalization of a partial action, as well as, we establish a connection between partial and global smash products via the construction of a surjective Morita context. In particular, in the case that the globalization is unital, these smash products are Morita equivalent. We show that there is a bijective correspondence between globalizable partial groupoid actions and symmetric partial groupoid algebra actions, extending similar result for group actions [9]. Moreover, as an application we give a complete description of all partial actions of a weak Hopf algebra on its ground field, which suggests a method to construct more general examples.
Let α and β be automorphisms on a ring R and Δ={δ i } i=0 n an (α,β)-higher derivation of order n... more Let α and β be automorphisms on a ring R and Δ={δ i } i=0 n an (α,β)-higher derivation of order n on R. It is shown that if 𝔉 is a right Gabriel filter on R that is α and β-invariant then 𝔉 is Δ-invariant. This not only extends a result due to C. Lomp and J. van den Berg [J. Pure Appl. Algebra 213, No. 4, 476-478 (2009; Zbl 1176.16022)] but also shows that every hereditary torsion theory on the category of right R-modules is higher differential in the sense of P. E. Bland [Int. J. Math. Math. Sci. 2005, No. 15, 2373-2387 (2005; Zbl 1100.16027)]. Also, we answer some questions posed by L. Vaš [in Int. J. Algebra 2, No. 13-16, 711-731 (2008; Zbl 1181.16027) and Commun. Algebra 37, No. 3, 794-810 (2009; Zbl 1226.16021)]. These results were independently obtained L. Vaš and C. Papachristou [in Trends in Mathematics, 165-174 (2010; Zbl 1200.16044)].
This is a short survey of Miguel Ferrero’s academic activity written on the occasion of his 70th ... more This is a short survey of Miguel Ferrero’s academic activity written on the occasion of his 70th birthday.
Contemporary Mathematics, 2009
The purpose of this paper is to develop the theory of non-commutative φ-rings using concepts from... more The purpose of this paper is to develop the theory of non-commutative φ-rings using concepts from distributive rings and rings with comparability. We recover the main statements for commutative φ-rings as obtained by Anderson and Badawi.
Results in Mathematics, 2003
ABSTRACT In this paper we study ω-distributive modules, where ω is a cardinal number. We extend a... more ABSTRACT In this paper we study ω-distributive modules, where ω is a cardinal number. We extend a characterization of distributive modules to ω-distributive modules. In particular, the case in which ω = n is a finite cardinal is considered. We apply the results to the case n = 2, obtaining new characterizations for distributive modules and rings. Special attention is given to saturated submodules and ideals.
Journal of Pure and Applied Algebra, 2007
We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgeb... more We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D * , and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D [[x]].
Journal of Algebra and Its Applications, 2011
ABSTRACT Let α be a partial action, having globalization, of a finite group G on a ring R with 1 ... more ABSTRACT Let α be a partial action, having globalization, of a finite group G on a ring R with 1 as defined by M. Dokuchaev, M. Ferrero and A. Paques, [J. Pure Appl. Algebra 208, No. 1, 77-87 (2007; Zbl 1142.13005)], where α=([D g ],[α g ] g∈G ), D g is an ideal with a central idempotent 1 g of R and α g :D g -1 →D g is an isomorphism of rings. Let R α ={r∈R∣α g (r1 g -1 )=r1 g , for all g∈G}. Then R is called a partial Galois extension of R α if there exists x i ,y i ∈R ∑ i x i α g (y i 1 g -1 )=δ 1,g 1 R for every g∈G. Denote the centralizer of a subset X of R in R by C R (X) and C R (R) by C(R). Suppose R is an α-partial Galois Azumaya extension of R α , that is, R is an α-partial Galois extension of R α which is an Azumaya algebra over C(R) α . Then the authors show some one to one correspondences among sets of suitable separable subalgebras of R, R α and C R (R α ). These results extend similar results for Galois extensions of an Azumaya algebra due to F. R. DeMeyer, R. Alfaro and G. Szeto.
Journal of Algebra, 2005
Right chain semigroups are semigroups in which right ideals are linearly ordered by inclusion. Mu... more Right chain semigroups are semigroups in which right ideals are linearly ordered by inclusion. Multiplicative semigroups of right chain rings, right cones, right invariant right holoids and right valuation semigroups are examples. The ideal theory of right chain semigroups is described in terms of prime and completely prime ideals, and a classification of prime segments is given, extending to these semigroups results on right cones proved by Brungs and Törner [H.H. Brungs, G. Törner, Ideal theory of right cones and associated rings, J. Algebra 210 (1998) 145-164].
Communications in Algebra, 2010
In this article, we discuss necessary and sufficient conditions for the crossed product S = R★αG ... more In this article, we discuss necessary and sufficient conditions for the crossed product S = R★αG by a twisted partial action α of a finite group G on a ring R to be separable over its center.
Canadian Mathematical Bulletin, 1999
Journal of Pure and Applied Algebra
We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgeb... more We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive and infinite dimensional chain coalgebras turn out to be finite duals of left noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be se...
Publicationes mathematicae
In this paper we introduce the notions of partial bimodule algebra and partial bico- module algeb... more In this paper we introduce the notions of partial bimodule algebra and partial bico- module algebra. We also deal with the existence of globalizations for these structures, generalizing related results appeared in [2, 4]. As an application we construct the partial (L;R)-smash product, extending the corresponding global notion appeared in [14] to the context of partial Hopf actions.
Journal of Pure and Applied Algebra, 2015
ABSTRACT In this paper we introduce the notion of partial action of a weak Hopf algebra on algebr... more ABSTRACT In this paper we introduce the notion of partial action of a weak Hopf algebra on algebras, unifying the notions of partial group action [11], partial Hopf action ([2],[3],[9]) and partial groupoid action [4]. We construct the fundamental tools to develop this new subject, namely, the partial smash product and the globalization of a partial action, as well as, we establish a connection between partial and global smash products via the construction of a surjective Morita context. In particular, in the case that the globalization is unital, these smash products are Morita equivalent. We show that there is a bijective correspondence between globalizable partial groupoid actions and symmetric partial groupoid algebra actions, extending similar result for group actions [9]. Moreover, as an application we give a complete description of all partial actions of a weak Hopf algebra on its ground field, which suggests a method to construct more general examples.
Let α and β be automorphisms on a ring R and Δ={δ i } i=0 n an (α,β)-higher derivation of order n... more Let α and β be automorphisms on a ring R and Δ={δ i } i=0 n an (α,β)-higher derivation of order n on R. It is shown that if 𝔉 is a right Gabriel filter on R that is α and β-invariant then 𝔉 is Δ-invariant. This not only extends a result due to C. Lomp and J. van den Berg [J. Pure Appl. Algebra 213, No. 4, 476-478 (2009; Zbl 1176.16022)] but also shows that every hereditary torsion theory on the category of right R-modules is higher differential in the sense of P. E. Bland [Int. J. Math. Math. Sci. 2005, No. 15, 2373-2387 (2005; Zbl 1100.16027)]. Also, we answer some questions posed by L. Vaš [in Int. J. Algebra 2, No. 13-16, 711-731 (2008; Zbl 1181.16027) and Commun. Algebra 37, No. 3, 794-810 (2009; Zbl 1226.16021)]. These results were independently obtained L. Vaš and C. Papachristou [in Trends in Mathematics, 165-174 (2010; Zbl 1200.16044)].
This is a short survey of Miguel Ferrero’s academic activity written on the occasion of his 70th ... more This is a short survey of Miguel Ferrero’s academic activity written on the occasion of his 70th birthday.
Contemporary Mathematics, 2009
The purpose of this paper is to develop the theory of non-commutative φ-rings using concepts from... more The purpose of this paper is to develop the theory of non-commutative φ-rings using concepts from distributive rings and rings with comparability. We recover the main statements for commutative φ-rings as obtained by Anderson and Badawi.
Results in Mathematics, 2003
ABSTRACT In this paper we study ω-distributive modules, where ω is a cardinal number. We extend a... more ABSTRACT In this paper we study ω-distributive modules, where ω is a cardinal number. We extend a characterization of distributive modules to ω-distributive modules. In particular, the case in which ω = n is a finite cardinal is considered. We apply the results to the case n = 2, obtaining new characterizations for distributive modules and rings. Special attention is given to saturated submodules and ideals.
Journal of Pure and Applied Algebra, 2007
We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgeb... more We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D * , and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D [[x]].
Journal of Algebra and Its Applications, 2011
ABSTRACT Let α be a partial action, having globalization, of a finite group G on a ring R with 1 ... more ABSTRACT Let α be a partial action, having globalization, of a finite group G on a ring R with 1 as defined by M. Dokuchaev, M. Ferrero and A. Paques, [J. Pure Appl. Algebra 208, No. 1, 77-87 (2007; Zbl 1142.13005)], where α=([D g ],[α g ] g∈G ), D g is an ideal with a central idempotent 1 g of R and α g :D g -1 →D g is an isomorphism of rings. Let R α ={r∈R∣α g (r1 g -1 )=r1 g , for all g∈G}. Then R is called a partial Galois extension of R α if there exists x i ,y i ∈R ∑ i x i α g (y i 1 g -1 )=δ 1,g 1 R for every g∈G. Denote the centralizer of a subset X of R in R by C R (X) and C R (R) by C(R). Suppose R is an α-partial Galois Azumaya extension of R α , that is, R is an α-partial Galois extension of R α which is an Azumaya algebra over C(R) α . Then the authors show some one to one correspondences among sets of suitable separable subalgebras of R, R α and C R (R α ). These results extend similar results for Galois extensions of an Azumaya algebra due to F. R. DeMeyer, R. Alfaro and G. Szeto.
Journal of Algebra, 2005
Right chain semigroups are semigroups in which right ideals are linearly ordered by inclusion. Mu... more Right chain semigroups are semigroups in which right ideals are linearly ordered by inclusion. Multiplicative semigroups of right chain rings, right cones, right invariant right holoids and right valuation semigroups are examples. The ideal theory of right chain semigroups is described in terms of prime and completely prime ideals, and a classification of prime segments is given, extending to these semigroups results on right cones proved by Brungs and Törner [H.H. Brungs, G. Törner, Ideal theory of right cones and associated rings, J. Algebra 210 (1998) 145-164].
Communications in Algebra, 2010
In this article, we discuss necessary and sufficient conditions for the crossed product S = R★αG ... more In this article, we discuss necessary and sufficient conditions for the crossed product S = R★αG by a twisted partial action α of a finite group G on a ring R to be separable over its center.
Canadian Mathematical Bulletin, 1999
Journal of Pure and Applied Algebra
We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgeb... more We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive and infinite dimensional chain coalgebras turn out to be finite duals of left noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be se...