Presentations of trivial extensions of finite dimensional algebras and a theorem of Sheila Brenner

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. We describe the ordinary quiver and relations for T(Λ) = Λ ⋉ D(Λ), the trivial extension of Λ by its minimal injective cogenerator D(Λ), and also for the repetitive algebra Λ of Λ. Associated with this description we give an application of a theorem of Sheila Brenner.Fil: Fernández, Elsa Adriana. Universidad Nacional de la Patagonia. Facultad de Ingeniería. Sede Puerto Madryn; ArgentinaFil: Platzeck, Maria Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin