Yet another way to deal with the multiplication of $\Lambda(A)$

PROPOSITION 06.2   $\Lambda(A)$ is generated by $\{(1-c T^n)_W ; c\in A, n\in \mathbb{N}\}$ as a topological additve group.

PROOF.. Induction. (We leave it as Exercise 6.1) $\qedsymbol$

PROPOSITION 06.3   Let $a,b\in A$. Assume $n,m \in\mathbb{Z}_{>0}$ such that $\gcd(n,m)=d$, $n=n_1d, m=n_2 d$. Then:

$$(1-a T^n)_W(1- b T^m)_W= (1-a^{m_1} b^{n_1} T^{n_1 m_1 d})^{d}$$

(Exercise 6.2)* Note: The answer can be somewhat different than that in the statement. Sorry about that.

COROLLARY 06.4   The multiplication of $\Lambda(A)$ surely remain in $\Lambda(A)$ as it should be.