# Question concerning modules over a Clifford algebra

Let $R$be a commutative ring with unit element, and $M$ be an $R-$module. Let $f:M \times M \to R$ be a nondegenerate symmetric bilinear quadratic form, and $C(f)$ be the corresponding Clifford algebra with $\mathbb{Z}_2$ grading $C(f)=C(f)^{0} \oplus C(f)^1$. Now let $P=P^0 \oplus P^1$ be a graded $C(f)$-module which is irreducible. Thus there is a natural multiplication map $\psi : C(f) \otimes P^0 \to P$. My question is :

Is this map necessarily an isomorphism ?

I am able to see that this map is epimorphic (using the irreducibility assumption), but not able to see why this map must be injective.

Edit :

Motivation for this problem comes from trying to prove proposition 12(6.3) of Fibre Bundles by Husemoller. Pic attached :

What's the motivation for the question? If you pick $R = M = k$ a field and $f = 0$, then $C := C(f) \cong k[\varepsilon]/(\varepsilon^2)$ with the ${\mathbb Z}/2{\mathbb Z}$-grading coming from the natural $\mathbb Z$-grading. Then the residue field of $C$ is an irreducible graded $C$-module for which $\psi$ is not an isomorphism.
• Thanks, I think you are right. I have updated the question so as to include the motivation. I think Husemoller is assuming $f$ to be non-degenerate somewhere. I have accordingly added this condition. Could you advise some example with non-degenerate $f$ ? – user90041 May 28 '16 at 18:12