Azumaya algebras with involution, polarizations, and linear generalized identities (Q1906629)

The author shows that the existence of certain linear generalized identities (GIs) for a ring \(R\) with involution * and center \(C\), forces \(R\) to be Azumaya. The statements here are somewhat less general than those in the paper. The first theorem strengthens one of \textit{A. Braun} [J. Algebra 109, 166-171 (1987; Zbl 0621.16005)] by showing that when \(X^* - \sum a_i X b_i\) is a GI for \(R\), then \(R\) is Azumaya, so this conclusion holds if the canonical map from \(R \otimes_C R^{\text{op}}\) to \(\text{End}_C R\) is onto. Next the author gives examples showing that GIs of the form \(X - \sum (a_i Xb_i + b_i^* X^* a_i^*)c_i\) or \(X -\sum(a_i X + X^* a^*_i)b_i\), where \(a_i, b_i \in R\) and \(c_i \in C\), do not force \(R\) to be Azumaya. However, if \(X^* - \sum(a_i X + X^* a^*_i)b_i\) is a GI for \(R\), then: i) \(R\) is Azumaya and * is of the first kind if \(1 - \sum a_i^* b_i\) is invertible; ii) \(R\) is Azumaya and * is of the first kind if \(R\) is *-prime, unless \(\sum a_i Xb_i\) is a GI and \(\sum a^*_i b_i = 1\); and iii) if \(R\) is a semiprime ring then \(R\) is a direct sum \((R_0,^*) \oplus (R_1,^*)\) with \(\sum a_i Xb_i\) a GI of \(R_0\), \(1 - \sum a^*_i b_i \in R_1\), and \(R_1\) is Azumaya and a finite direct sum of prime Azumaya algebras with involutions of the first kind.