Hopfish structure and modules over irrational rotation algebras

Inspired by the group structure on $S^1\mathrm{/}\text{bbZ}$, we introduce a weak hopfish structure on an irrational rotation algebra $A$ of finite Fourier series. We consider a class of simple $A$-modules defined by invertible elements, and we compute the tensor product between these modules defined by the hopfish structure. This class of simple modules turns out to generate an interesting commutative unital ring.