  The purpose of this work is to generalize part of the theory behind Faugere's
"F5" algorithm. This is one of the fastest known algorithms to compute a
Groebner basis of a polynomial ideal I generated by polynomials
f_{1},...,f_{m}. A major reason for this is what Faugere called the algorithm's
"new" criterion, and we call "the F5 criterion"; it provides a sufficient
condition for a set of polynomials G to be a Groebner basis. However, the F5
algorithm is difficult to grasp, and there are unresolved questions regarding
its termination.
  This paper introduces some new concepts that place the criterion in a more
general setting: S-Groebner bases and primitive S-irreducible polynomials. We
use these to propose a new, simple algorithm based on a revised F5 criterion.
The new concepts also enable us to remove various restrictions, such as proving
termination without the requirement that f_{1},...,f_{m} be a regular sequence.
