Exercise 1, § 3

Let $\Delta$ be a commutative monoid written additively with the identity element denoted by 0, all of whose elements are cancellable. Let E be a graded A-algebra of type $\Delta$ and suppose that E admits a unit element $e = \sum_{\alpha \in \Delta} e_\alpha$ where $e_\alpha \in E_\alpha$. Show that necessarily $e = e_0 \in E_0$.