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Eichler's trace formula for traces of the Brandt-Eichler matrices is proved for arbitrary totally definite orders in central simple algebras of prime index over global fields. A formula for type numbers of such orders is proved as an application.

The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let $K$ be a unital commutative ring, not necessarily a field. Given a unital $K$-algebra $S$, where $K$ is contained in the center of $S$, $n\in \mathbb{N}$, the goal of this paper is to study the question: when can a homomorphism $\phi :{\mathrm{M}}_{n}\left(K\right)\to {\mathrm{M}}_{n}\left(S\right)$ be extended to an inner automorphism of ${\mathrm{M}}_{n}\left(S\right)$? As an application of main results presented in the paper, it is proved that if $S$ is a semilocal...

Let $D$ be a division ring finite dimensional over its center $F$. The goal of this paper is to prove that for any positive integer $n$ there exists $a\in {D}^{\left(n\right)},$ the $n$th multiplicative derived subgroup such that $F\left(a\right)$ is a maximal subfield of $D$. We also show that a single depth-$n$ iterated additive commutator would generate a maximal subfield of $D.$

Let $M$ be a family of Mumford-type, that is, a family of polarized complex abelian fourfolds as introduced by Mumford in [9]. This family is defined starting from a quaternion algebra $A$ over a real cubic number field and imposing a condition to the corestriction of such $A$. In this paper, under some extra conditions on the algebra $A$, we make this condition explicit and in this way we are able to describe the polarization and the complex structures of the fibers. Then, we look at the non simple $CM$-fibers...