Centralizers in Domains of Gelfandkirillov Dimension 2
نویسندگان
چکیده
Given an affine domain of Gelfand–Kirillov dimension 2 over an algebraically closed field, it is shown that the centralizer of any non-scalar element of this domain is a commutative domain of Gelfand–Kirillov dimension 1 whenever the domain is not polynomial identity. It is shown that the maximal subfields of the quotient division ring of a finitely graded Goldie algebra of Gelfand– Kirillov dimension 2 over a field F all have transcendence degree 1 over F . Finally, centralizers of elements in a finitely graded Goldie domain of Gelfand–Kirillov dimension 2 over an algebraically closed field are considered. In this case, it is shown that the centralizer of a non-scalar element is an affine commutative domain of Gelfand–Kirillov dimension 1.
منابع مشابه
Noetherian Hopf Algebra Domains of Gelfand-kirillov Dimension Two
We classify all noetherian Hopf algebras H over an algebraically closed field k of characteristic zero which are integral domains of GelfandKirillov dimension two and satisfy the condition ExtH(k, k) 6= 0. The latter condition is conjecturally redundant, as no examples are known (among noetherian Hopf algebra domains of GK-dimension two) where it fails.
متن کاملCentralizers in Domains of Finite Gelfand-kirillov Dimension
We study centralizers of elements in domains. We generalize a result of the author and Small [4], showing that if A is a finitely generated noetherian domain and a ∈ A is not algebraic over the extended centre of A then the centralizer of a has Gelfand-Kirillov dimension at most one less than the Gelfand-Kirillov dimension of A. In the case that A is a finitely generated noetherian domain of GK...
متن کاملUnruffled Extensions and Flatness over Central Subalgebras
A condition on an affine central subalgebra Z of a noetherian algebra A of finite GelfandKirillov dimension, which we call here unruffledness, is shown to be equivalent in some circumstances to the flatness of A as a Z-module. Unruffledness was studied by Borho and Joseph in work on enveloping algebras of complex semisimple Lie algebras, and we discuss applications of our result to enveloping a...
متن کاملSemistar dimension of polynomial rings and Prufer-like domains
Let $D$ be an integral domain and $star$ a semistar operation stable and of finite type on it. We define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong S-domains. As an application, we give new characterizations of $star$-quasi-Pr"{u}fer domains and UM$t$ domains in terms of dimension inequal...
متن کاملOn solubility of groups with finitely many centralizers
For any group G, let C(G) denote the set of centralizers of G.We say that a group G has n centralizers (G is a Cn-group) if |C(G)| = n.In this note, we prove that every finite Cn-group with n ≤ 21 is soluble andthis estimate is sharp. Moreover, we prove that every finite Cn-group with|G| < 30n+1519 is non-nilpotent soluble. This result gives a partial answer to aconjecture raised by A. Ashrafi in ...
متن کامل