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Abstract: We study del Pezzo surfaces of degree 1 of the form in the weighted projective space , where k is a perfect field of characteristic not 2 or 3 and . Over a number field, we exhibit an infinite family of (minimal) counterexamples to weak approximation amongst these surfaces, via a Brauer–Manin obstruction. [Copyright &y& Elsevier]

Abstract: We develop a rank variety for finite-dimensional modules over a certain class of finite-dimensional local -algebras, . Included in this class are the truncated polynomial algebras , with an algebraically closed field and arbitrary. We prove that these varieties characterise projectivity of modules (Dade’s lemma) and examine the implications for the tree class of the stable Auslander–Reiten quiver. We also extend our rank varieties to infinitely generated modules and verify Dade’s lemma in this context. [Copyright &y& Elsevier]

The two main classes of finite dimensional composition algebras, Hurwitz and symmetric, are characterized in terms of the flexible identity, and even of the third power associative identity, over fields of arbitrary characteristic. [ABSTRACT FROM AUTHOR]