Relation between two generators of a Grassman algebra

Question

Im considering the simple case of a Grassman algebra $\mathcal{A}$ defined by a set of generators $\{\theta_{i},\bar{\theta}_{i}\}$ with $i=1...N$, so that for any $i,j$: \begin{eqnarray} \left\{\theta_{i},\theta_{j}\right\}=\left\{ \bar{\theta}_{i},\bar{\theta}_{j} \right\}=\left\{ \theta_{i},\bar{\theta}_{j} \right\}=0, \end{eqnarray} where $\{a,b\}=ab+ba$. I am considering a case where a boundary condition between generators is found in the following way: \begin{eqnarray} \bar{\theta}_{1}\theta_{N}=c,\hspace{10pt} c\in\mathbb{C}, \end{eqnarray} that is, there is a relation between the generator $\bar{\theta}$ of index $1$ and the generator $\theta$ of index $N$, but NOT between any others in between, i.e. for any $i,j\in{2,...,N-1}$ the only condition is the one with brackets $\{\theta_{i},\theta_{j}\}$. What can be said in this case about the above condition? If I am not missing something, since the number $\bar{\theta}_{1}\theta_{N}\in \mathcal{A}$ is composed by a superposition of the generators, it must belong to the algebra; however, given that $c\in\mathbb{C}$ that seems hardly possible unless $c=0$. Is the above condition impossible then? Or otherwise, does it allow to make some statement about the behaviour of the $\{\bar{\theta},\theta\}$ in general?