Graded Frobenius algebras from tensor algebras of bimodules

We consider certain quotient algebras of tensor algebras of bimodules $M$ over a finite-dimensional algebra $R$, and we investigate Frobenius type properties of such algebras. Our main interest is in the case where $M=R^{\ast }$, the linear dual of $R$. We obtain a large class of Frobenius or symmetric algebras, which are also equipped with a finite grading.