## Abstract

We consider the following problem: Under what assumptions are one or more of the following equivalent for a ring R: (A) R is Morita equivalent to a ring with involution, (B) R is Morita equivalent to a ring with an anti-automorphism, (C) R is Morita equivalent to its opposite ring. The problem is motivated by a theorem of Saltman which roughly states that all conditions are equivalent for Azumaya algebras. Based on the recent general bilinear forms of [10], we present a general machinery to attack the problem, and use it to show that (C) (B) when R is semilocal or Q-finite. Further results of similar flavor are also obtained, for example: If R is a semilocal ring such that M_{R}(n) has an involution, then M_{R}(2) has an involution, and under further mild assumptions, R itself has an involution. In contrast to that, we demonstrate that (B) (A). Our methods also give a new perspective on the Knus-Parimala-Srinivas proof of Saltman's Theorem. Finally, we give a method to test Azumaya algebras of exponent 2 for the existence of involutions, and use it to construct explicit examples of such algebras.

Original language | English |
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Pages (from-to) | 26-61 |

Number of pages | 36 |

Journal | Journal of Algebra |

Volume | 430 |

DOIs | |

State | Published - 5 May 2015 |

Externally published | Yes |

### Bibliographical note

Funding Information:This research was supported by an ERC grant # 226135 and by the Lady Davis Fellowship Trust, Hebrew University of Jerusalem .

Publisher Copyright:

© 2015 Elsevier Inc..

## Keywords

- Anti-automorphism
- Azumaya algebra
- Bilinear form
- Brauer group
- Corestriction
- General bilinear form
- Involution
- Morita equivalence
- Semilocal ring

## ASJC Scopus subject areas

- Algebra and Number Theory