Let $A$ be a Frobenius algebra (i.e. a finite-dimensional, unital, associative algebra equipped with a non-degenerate bilinear form). We define an operation on the ideals of $A$ as follows:
$$I \cdot J = (I \cap J) + I^{\bot},$$
where $I^{\bot}$ denotes the right annihilator of $I$. I have troubling proving the following statements for all ideals $I,J$ and $K$ of A:
1) $I + I^{\bot}= A$
2) $(I \cdot J) \cdot (I \cdot K)=(J \cdot I) \cdot (J \cdot K)$
Any help?