Conference abstracts

Session S05 - Rings and Algebras

July 29, 16:10 ~ 16:35

## Color involutions of primitive graded rings.

### Irina Sviridova

### University of Brasilia, Brazil - I.Sviridova@mat.unb.br

Kaplansky's Theorem [2] characterizes involutions of primitive rings with a nonzero socle in terms of hermitian and alternate forms. In 1997 M.L.Racine [3] constructed similar structure theory for primitive associative superalgebras. And Yu.A. Bakhturin, M. Bresar, M. Kochetov [1] obtained similar results for graded rings with graded involutions.

We present analogous characterizations of primitive graded rings in terms of twisted pairing. This implies the extension of Kaplansky's Theorem for primitive graded rings with a color involution in case of a grading by a cyclic group of a prime order. We also obtain some corollaries on color involutions of finite dimensional simple graded algebras. In particular, these results generalise the corresponding theorems of [2].

The work is partially supported by CNPq, CAPES.

[1] Yu.A. Bakhturin, M. Bresar, M. Kochetov, Group gradings on finitary simple Lie algebras, Int. J. Algebra Comp., 22(2012), 125-146.

[2] N. Jacobson, Structure of Rings, AMS Colloquium Publication 37, AMS, Providence, R.I., 1964.

[2] M.L. Racine, Primitive Superalgebras with Superinvolution, J. Algebra 206(2)(1998), 588-614.

Joint work with Keidna Cristiane Oliveira Souza (University of Brasilia, Brazil).