Distributive Algebra w/o Additive Inverse

Distributive Algebra w/o Additive Inverse

Suppose we are working in an algebra $\left( X, +, \ast \right)$. Both
binary operators are associative and commutative, and $\ast$ distributes
over $+$. However, not all of the elements have an additive inverse (e.g.
there is no $B \neq 0$ such that $A+B=0$).
Is this a sufficient condition to show that all elements (except for $0,
1$) cannot have a multiplicative inverse? If so, how do I go about proving
it? I haven't found any counterexamples so far, and my attempted proofs
are going nowhere.