We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient condit...We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient conditions for GKdim(A)=GKdim(D)+1 are given.In particular,we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates,i.e.,GKdim(A)is either 3 or∞in this case.Our results generalize several existing results in the literature and can be applied to determine the growth,GK-dimension,simplicity and cancellation properties of some GWAs.展开更多
In this paper, by using Gr bner-Shirshov bases theories, we prove that each countably generated associative differential algebra （resp., associative λ-algebra, associa- tive Ω-differential algebra） can be embedded...In this paper, by using Gr bner-Shirshov bases theories, we prove that each countably generated associative differential algebra （resp., associative λ-algebra, associa- tive Ω-differential algebra） can be embedded into a simple 2-generated associative differ- ential algebra （resp., associative Ωalgebra, associative λ-differential algebra）.展开更多
Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension...Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method： Grobner-Shirshov basis method. We develop the GrSbner-Shirshov basis theory of differential difference al- gebras, and of finitely generated modules over differential difference algebras, respectively. Then, via GrSbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.展开更多
Let R be a ring with identity 1. Jacobson＇s lemma states that for any a, b ∈ R, if 1 - ab is invertible then so is 1 - ba. Jacobson＇s lemma has suitable analogues for several types of generalized inverses, e.g., Dr...Let R be a ring with identity 1. Jacobson＇s lemma states that for any a, b ∈ R, if 1 - ab is invertible then so is 1 - ba. Jacobson＇s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Grobner-Shirshov basis theory to obtain the inverse of 1 - ab in terms of （1 - ba）-1, assuming the latter exists.展开更多
基金supported by Huizhou University(Grant Nos.hzu202001 and 2021JB022)the Guangdong Provincial Department of Education(Grant Nos.2020KTSCX145 and 2021ZDJS080)。
文摘We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient conditions for GKdim(A)=GKdim(D)+1 are given.In particular,we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates,i.e.,GKdim(A)is either 3 or∞in this case.Our results generalize several existing results in the literature and can be applied to determine the growth,GK-dimension,simplicity and cancellation properties of some GWAs.
文摘In this paper, by using Gr bner-Shirshov bases theories, we prove that each countably generated associative differential algebra （resp., associative λ-algebra, associa- tive Ω-differential algebra） can be embedded into a simple 2-generated associative differ- ential algebra （resp., associative Ωalgebra, associative λ-differential algebra）.
文摘Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method： Grobner-Shirshov basis method. We develop the GrSbner-Shirshov basis theory of differential difference al- gebras, and of finitely generated modules over differential difference algebras, respectively. Then, via GrSbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.
文摘Let R be a ring with identity 1. Jacobson＇s lemma states that for any a, b ∈ R, if 1 - ab is invertible then so is 1 - ba. Jacobson＇s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Grobner-Shirshov basis theory to obtain the inverse of 1 - ab in terms of （1 - ba）-1, assuming the latter exists.
