Uriya A. First

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20242026

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math.RA2025

Counterexamples in Involutions of Azumaya Algebras

Uriya First, Ben Williams

Suppose $A$ is an Azumaya algebra over a ring $R$ and $I ¨ $ is an involution of $A$ extending an order- $2$ automorphism $I^» : R \to R$ . We say $I ¨ $ is extraordinary if there does not e…

math.RA2025

Irredundant Generating Sets for Matrix Algebras

Yonatan Blumenthal, Uriya First

Let $F$ be a field. We show that the largest irredundant generating sets for the algebra of $n \times n$ matrices over $F$ have $2 n - 1$ elements when $n > 1$ . (A result of Laffey stat…

math.RA2024

Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

Uriya A. First

Let $(A, I ¨ )$ be an Azumaya algebra with orthogonal involution over a ring $R$ with $2 \in R^{\times}$ . We show that if $(A, I ¨ )$ admits an improper isometry, i.e., an element $a \in A$ …

math.RA2024

Spaces of Generators for Azumaya Algebras with Unitary Involution

Omer Cantor, Uriya A. First

Let $A$ be a finite dimensional algebra (possibly with some extra structure) over an infinite field $K$ and let $r \in N$ . The $r$ -tuples $(a_{1}, \dots, a_{r}) \in A^{r}$ which fail…