Examples in finite Gel'fand-Kirillov dimension
The author modifies constructions of \textit{K. I. Beidar} [Usp. Mat. Nauk 36, No. 6(222), 203-204 (1981; Zbl 0482.16008)] and L. Small and proves that for countably generated prime \(F\)-algebras of finite GK-dimension there exists an affinization having finite GK-dimension. Using this result, the author constructs the following examples. (1) For any field there exists a prime affine algebra of GK-dimension two that is neither primitive nor PI. (2) For any countable field \(F\) there exists a prime affine \(F\)-algebra of GK-dimension three that has non-nil Jacobson radical. (3) For any countable field \(F\) there exists an affine primitive \(F\)-algebra of GK-dimension at most four with center equal to a polynomial ring. (4) For a countable field \(F\) there exists a primitive affine Jacobson \(F\)-algebra of GK-dimension three that does not satisfy the Nullstellensatz.
scientific article; zbMATH DE number 1924687
scientific article