If $R$ is just a ring, then $R((t))$, the ring of formal Laurent series with coefficients in $R$, is given by elements of the form $\sum r_kt^k$ such that $r_k=0$ for $k$ smaller than some constant. Giving $R$ the discrete topology this can be rewritten to $\lim_{k\to-\infty}r_k=0$.
My question is: If $R$ is already a topological ring, what is the most relaxed condition we can put on the "negative coefficients" to make this still into a well defined ring? Here the product should be the usual Cauchy product $$\sum_ia_it^i\sum_jb_jt^j:=\sum_kt^k\sum_{i+j=k}a_ib_j$$
So far I could proof one would need to have that $\sum_{k>0}a_kr_{-k}$ converges for arbitrary $a_k$ so that $R[[t]]\subset R((t))$ (which should definietly hold). Then the product of any two of these is defined and it is commutative. But I don't know if this is enough to have an associative product?