A non-commutative F5F5 algorithm with an application to the computation of Loewy layers

We provide a non-commutative version of the F5F5 algorithm, namely for right-modules over path algebra quotients. It terminates, if the path algebra quotient is a basic algebra. We show that the signatures used in the F5F5 algorithm allow to read off a basis for each Loewy layer, provided that a negative degree monomial ordering is used. As a byproduct, Gröbner bases in this setting can be computed more efficiently with the F5F5 algorithm than with Buchberger's algorithm.