= Solution
Let $g$ be a <p-regular element>. Its eigenvalues on $M$ are roots of unity of order prime to $p$. If $\widehat\lambda_1,\ldots,\widehat\lambda_d$ are their <Teichmuller lifts> to characteristic-zero roots of unity, the <Brauer character> is
$$
\boxed{\chi_M(g)=\sum_{r=1}^d\widehat\lambda_r.}
$$
It depends only on the conjugacy class of $g$, is additive in short exact sequences, and equals the restriction of an ordinary character whenever the representation lifts.
For linear independence, choose a splitting <p-modular system> and let $P_i$ be the <projective cover> of the simple module $S_i$. A projective lattice lifting $P_i$ has an ordinary character $\Phi_i$ that vanishes on p-singular elements. Reduction and ordinary character orthogonality give
$$
\frac1{|G|}\sum_{g\ p\text{-regular}}
\Phi_i(g^{-1})\chi_{S_j}(g)
=\dim\operatorname{Hom}_{kG}(P_i,S_j)
=\delta_{ij},
$$
because $S_i$ is the head of $P_i$. Pairing a relation $\sum_jc_j\chi_{S_j}=0$ with every $\Phi_i$ yields $c_i=0$ for every $i$. Hence
$$
\boxed{\{\chi_{S_i}\}\text{ is linearly independent over }\mathbb C.}
$$
This is the linear-independence part of the <Brauer–Nesbitt theorem>.
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