Let $C(l)$ be a cyclic ring with elements $0,e,2e,...,(n-1)e$. Addition is defined by adding coefficients. Multiplication is defined such that $e^2=le$. So if we multiply two elements together, say $xe$ and $ye$, we get $xe*ye=xye^2=xyle$. Let $C(f)$ be the same ring except with $e^2=fe$.
If $C(l)$ is infinite, can $C(l)$ ever be isomorphic to $C(f)$?
If $C(l)$ is finite, what is the condition on $l$ and $f$ in order that there be an isomorphism?