Open Access𝐺-graded central polynomials and 𝐺-graded Posner’s theorem
Author(s) -
Yakov Karasik
Publication year - 2019
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/7736
Subject(s) - algorithm , annotation , type (biology) , computer science , artificial intelligence , mathematics , biology , ecology
Let F \mathbb {F} be a characteristic zero field, let G G be a residually finite group, and let W W be a G G -prime and polynomial identity F \mathbb {F} -algebra. By constructing G G -graded central polynomials for W W , we prove the G G -graded version of Posner’s theorem. More precisely, if S S denotes all nonzero degree e e central elements of W W , the algebra S − 1 W S^{-1}W is G G -graded simple and finite dimensional over its center. Furthermore, we show how to use this theorem in order to recapture a result of Aljadeff and Haile stating that two G G -simple algebras of finite dimension are isomorphix if and only if their ideals of graded identities coincide.