## Abstract

We use the generalized correspondence to show that there is a canonical set isomorphism Inn(R) \Aut^{−}(R) ≅ Inn(M_{n}(R))\Aut^{−}(M_{n}(R)), provided R_{R} is the only right R-module N satisfying N^{n} ≅R^{n}, and also to prove a variant of a theorem of Osborn. Namely, we classify all semisimple rings with involution admitting no non-trivial idempotents that are invariant under the involution.

We introduce the new notion of general bilinear forms (generalizing sesquilinear forms) and prove that for every ring R (not necessarily commutative, possibly without involution) and every right R-module M which is a generator (i.e., R_{R} is a summand of M^{n} for some n ∈ ℕ), there is a one-to-one correspondence between the anti-automorphisms of End(M) and the general regular bilinear forms on M, considered up to similarity. This generalizes a well-known similar correspondence in the case R is a field. We also demonstrate that there is no such correspondence for arbitrary R-modules.

Original language | English |
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Pages (from-to) | 145-183 |

Number of pages | 39 |

Journal | Israel Journal of Mathematics |

Volume | 205 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2015 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2015, Hebrew University of Jerusalem.

## ASJC Scopus subject areas

- General Mathematics