Let S=K[x1,...,xn] or S=K[[x1,...,xn]] be either a
polynomial or a formal power series ring in a finite number of variables
over a field K of characteristic p>0 with [K:Kp]<∞. Let R be the hypersurface S/fS where f is a nonzero nonunit element of S. If e is a positive integer, F∗e(R) denotes the R-algebra structure induced on R via the e-times iterated Frobenius map ( r→rpe ). We describe a matrix factorizations of f whose cokernel is isomorphic to F∗e(R) as R-module. The presentation of F∗e(R) as the cokernel of a matrix factorization of f enables us to find a characterization from which we can decide when the ring S[[u,v]]/(f+uv) has finite F-representation type (FFRT) where S=K[[x1,...,xn]]. This allows us to create a class of rings that have finite F-representation type but not finite CM type. For S=K[[x1,...,xn]], we use this presentation to show that the ring S[[y]]/(ypd+f) has finite F-representation type for any f in S. Furthermore, we prove that S/I has finite F-representation type when I is a monomial ideal in either S=K[x1,...,xn] or S=K[[x1,...,xn]]. Finally, this presentation enables us to compute the F-signature of the rings S[[u,v]]/(f+uv) and S[[z]]/(f+z2) where S=K[[x1,...,xn]] and f is a monomial in the ring S. When R is a Noetherian ring of prime characteristic that has FFRT, we prove that R[x1,...,xn] and R[[x1,...,xn]] have FFRT.
We prove also that over local ring of prime characteristic a module has FFRT if and only it has FFRT by a FFRT system. This enables us to show that if M is a finitely generated module over Noetherian ring R of prime characteristic p, then the set of all prime ideals Q such that MQ has FFRT over RQ is an open set in the Zariski topology on \Spec(R)