= Solution
For the contrapositive of $(b)\Rightarrow(c)$, suppose $\operatorname{char}k=p>0$. Consider
$$
M=k[x_1,\ldots,x_n]/(x_1^p,\ldots,x_n^p).
$$
Let $x_i$ act by multiplication and $D_i$ by formal differentiation. Differentiation preserves the defining <ideal>, since $D_j(x_i^pf)=x_i^pD_jf$ in characteristic $p$. The relations $[D_i,x_j]=\delta_{ij}$ hold before quotienting and hence hold on $M$. Its <basis> consists of the monomials $x^\alpha$ with $0\le\alpha_i<p$, so \b[$\dim_kM=p^n>0$]. This is the explicit finite-dimensional module for the <Weyl algebra in positive characteristic>.
Therefore condition (b) fails in positive characteristic and implies characteristic zero. Combining this with the preceding two implications proves \b[$(a)\Longleftrightarrow(b)\Longleftrightarrow(c)$]. The same representation also has nonzero proper kernel, since the infinite-dimensional <Weyl algebra> cannot inject into a finite-dimensional endomorphism <algebra>, confirming its failure of simplicity in positive characteristic.
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