The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic p > 2 little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic; we showed that the so-called Tensor Product Theorem cannot be extended for infinite fields of positive characteristic p > 2. Furthermore we studied the Gelfand-Kirillov dimension of the relatively free algebras of verbally prime and related algebras. In this paper we compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras M a,a (E) ⊗ E and M 2a (E) are not PI equivalent in characteristic p > 2. Furthermore we show that the following algebras are not PI equivalent in positive characteristic: M a,b (E) ⊗ M c,d (E) and M ac+bd,ad+cb (E); and M a,b (E) ⊗ M c,d (E) and M e, f (E) ⊗ M g,h (E) when a ≥ b, c ≥ d, e ≥ f , g ≥ h, ac + bd = eg + f h, ad + bc = eh + f g and ac = eg. Here E stands for the infinite dimensional Grassmann algebra with 1, and M a,b (E) is the subalgebra of M a+b (E) of the block matrices with blocks a × a and b × b on the main diagonal with entries from E 0 , and off-diagonal entries from E 1 ; E = E 0 ⊕ E 1 is the natural grading on E.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.