Let $A= \bigoplus_{i=0}^{n}A_i$ be a finite dimensional algebra over a field $\mathbb{k}$ such that $A_0 \cong \mathbb{k} \cong A_n$. Consider the bilinear form $$\varphi: A_i \times A_{n-i} \to A_n$$ given by ordinary multiplication in $A$. We say $A$ is a Poincaré duality algebra if $\varphi(r,s)=0$ for all $s \in A_{n-i}$ implies $r=0$, for all $i \in \{0,1,...,n\}$.
I am looking for an example of a Poincare duality algebra such that for some $i, \; j \in \{0,1,...,n\}$ such that $i+j \leq n$ and $0 \neq r \in A_i$, $0 \neq s \in A_j$ we have $rs=0$. Any explicit example of such an algebra would be very helpful!
Let $V$ be any vector space equipped with a nondegenerate symmetric bilinear form $(\cdot,\cdot):V\times V\to \mathbb{k}$. Then you can form an algebra with $A_0=\mathbb{k}$, $A_1=V$, $A_2=\mathbb{k}$, $A_i=0$ for $i>2$, and multiplication of elements of $A_1$ given by $vw=(v,w)\in A_2$. Since the bilinear form is nondegenerate, this is a Poincare duality algebra. But if $\dim V>1$, then for any $v\in V$ the functional $(v,\cdot)$ must have nontrivial kernel, so there exists a nonzero $w$ such that $(v,w)=0$.