Primitive algebras with arbitrary Gelfand-Kirillov dimension
Primitive algebras of arbitrary Gelfand-Kirillov dimension \(\geq 2\) were first constructed by \textit{R. S. Irving} and \textit{R.~B.~Warfield}, jun. [J. Lond. Math. Soc., II. Ser. 36, 219-228 (1987; Zbl 0638.16016)]. However, their examples required the base field \(k\) to have an infinite dimensional algebraic extension. In the paper under review, the author removes this restriction on \(k\). His examples are generalizations of the Morse algebra; they are monomial algebras, in fact, quotients of the free algebra \(k\langle x,y\rangle\) by ideals generated by carefully chosen sequences of monomials. The primitive algebras that arise have some very interesting additional features: their only two-sided ideals are the zero ideal together with the powers of the Jacobson radical; so although the Gelfand-Kirillov dimension of these examples can be any real number \(\geq 2\), their classical Krull dimension is always equal to one. Furthermore, all homomorphic images modulo nonzero ideals are finite dimensional, but the algebras are neither left nor right Noetherian.
scientific article; zbMATH DE number 1251299
scientific article