On generic $G$-graded Azumaya algebras
Eli Aljadeff, Yakov Karasik

TL;DR
This paper constructs a universal $G$-graded Azumaya algebra that encapsulates the identities of finite dimensional $G$-graded simple algebras over an algebraically closed field, providing a framework for their classification.
Contribution
It introduces a generic $G$-graded Azumaya algebra with a one-to-one ideal correspondence and characterizes $G$-graded simple algebras admitting a division algebra form.
Findings
Constructed a universal $G$-graded Azumaya algebra $$
Established a correspondence between graded ideals and ring ideals
Characterized algebras admitting a $G$-graded division algebra form
Abstract
Let be an algebraically closed field of characteristic zero and let be a finite group. Consider -graded simple algebras which are finite dimensional and -central over , i.e. . For any such algebra we construct a \textit{generic} -graded algebra which is \textit{Azumaya} in the following sense. \textit{Correspondence of ideals}: There is one to one correspondence between the -graded ideals of and the ideals of the ring , the -center of . \textit{Artin-Procesi condition}: satisfies the -graded identities of and no nonzero -graded homomorphic image of satisfies properly more identities. \textit{Generic}: If is a -graded algebra over a field then it is a specialization of along an ideal $\mathfrak{a} \in…
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