For a given semigroup $S$, the semigroup algebra over some field $F$ is the set of formal sum with the convolution product, and is denoted by $F[S]$.
If we built the direct sum $$ \mathbb R \oplus \mathbb H $$ of the two $\mathbb R$-algebras, with $\mathbb R \cdot \mathbb H = 0$, why is this not isomorphic to $\mathbb R[S]$ for some semigroup $S$.
Any hints how to show this?
I looked at other examples, like $\mathbb R \oplus \mathbb R \cong \mathbb R[S]$ with $S = \{ 1', 1 + 1' \}$ where $1'$ is the unit in the second direct summand. But I fail to find an argument that the above direct sum $\mathbb R \oplus \mathbb H$ is not of this form.