Graded Prime PI-Algebras

The graded Gelfand--Kirillov dimension of verbally prime algebras

“…, x g r k . We have the following theorem [5]: We find the Z n -graded Gelfand-Kirillov dimension for M n (F ) using a graded version of Balaba [2], of the well-known theorem of Posner (see, e.g. the book by Drensky and Formanek [10]).…”

“…the book by Drensky and Formanek [10]). For this purpose we introduce some definitions and results obtained by Balaba [2]. Let G be any group and A be a G-graded algebra with unity, and denote with the symbol h(A) the set of the G-homogeneous elements of A.…”

“…Recall that by Proposition 1 of [2], the localization A S of A over S, where S is a set of homogeneous elements of the centre Z(A) of A, is a G-graded PI-algebra of central quotients of A. An algebra Q(A) A is called the left (right)-graded algebra of quotients of A if:…”

“…In particular, we use the graded version of the theorem of Posner [2], to extend some results about the Gelfand-Kirillov dimension of PI-algebras to the graded case. Then we are able to adapt the work of Procesi [19] to find the Z n -graded Gelfand-Kirillov dimension of M n (F ).…”

𝐺-graded central polynomials and 𝐺-graded Posner’s theorem

“…Remark 1.4. In a (English translation of) paper of Balaba [4] it seems that the author have proved a version of theorem 1.3 for every G. However, the proof given there is valid only for finite abelian groups G. Indeed, the author starts his proof (Proposition 2) by using Corollary 4.5 from [5] which is valid for finite groups (it is false for infinite groups, see 4.6). The end of the proof makes sense only for abelian groups.…”