On a generalized version of the Nakayama conjecture

Abstract

Nakayama proposed a conjecture which is equivalent to the following: If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite dimensional algebra over a field and the dominant dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is infinite, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is self-injective. In this paper we study a generalized version of this conjecture.