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

Abstract

Let F be characteristic zero field, G a residually finite group and W a G-prime and PI F-algebra. By constructing G-graded central polynomials for W , we prove the G-graded version of Posner's theorem. More precisely, if S denotes all non-zero degree e central elements of W , the algebra S −1 W is G-graded simple and finite dimensional over its center.Furthermore, we show how to use this theorem in order to recapture the result of Aljadeff and Haile stating that two G-simple algebras of finite dimension are isomorphix iff their ideals of graded identities coincide.

Semisimple Algebras and PI-Invariants of Finite Dimensional Algebras

“…For the proof we shall need to insert suitable e-central polynomial in the full G-graded polynomials of A constructed above. We recall from [14] that every finite dimensional G-graded simple admits an e-central multilinear polynomial c A , that is a nonidentity of A, central and G-homogeneous of degree e. Furthermore, it follows from its construction, that the polynomial c A alternates on certain sets of variables of equal homogeneous degree of cardinality equal dim F (A g ), for every g ∈ G. For the proof of Theorem 4.8 we shall need e-central polynomials with some additional properties.…”

Identities and central polynomials for real graded division algebras

“…. , x m,g ′ m ) in Id G (A), where for each g ∈ G the number of indexes j such that g ′ j = g is at most 3, such that f lies in the T G -ideal generated by f ′ together with the polynomials in (9), (10), (13) and (12).…”

“…It is clear that, for this polynomial, the difference in (15) lies in the T G -ideal generated by the polynomials in (12) and (13). Now write…”

“…Let g, h 1 , h 2 , h 3 be elements of G such that β A (g, h i ) = i for i = 1, 2, 3. The polynomial (13) x 1,g x 2,h1 x 3,g x 4,h2 x 5,g x 6,h3 x 7,g + x 1,g x 3,g x 5,g x 7,g x 2,h1 x 4,h2 x 6,h3 is a graded identity for A.…”

“…Central polynomials play an important role in p.i. theory, we refer the reader to [12], [13] for more details and applications.…”