Corollary

Let $G_+$ be the set of positive elements in $G$. The mapping $M \mapsto a(M)$ is a bijection of the set of major subsets of $G_+$, onto the set of ideals of $A$.

As $A = a(G_+)$, $a(M) \subset A$ is equivalent to $M \subset G_+$.

For example the maximal ideal $m(A)$ is equal to $a(S)$, where $S$ denotes the set of strictly positive elements of $G$.