Problems on definition of genus with multiplicative sequence.
The Wikipedia article describes the general notion of a genus, allowing for homomorphisms to general rings, for instance, to ${\mathbb Z}/2$. The authors of the book you are reading, prefer to work over ${\mathbb Q}$, i.e. when $R$ is a ${\mathbb Q}$-algebra, eliminating examples such as the ring ${\mathbb Z}/2$. Of course, one can still consider ring-homomorphisms
$$
\Omega\to R,
$$
but (if one thinks in terms of ${\mathbb Q}$-algebras) it is more natural to consider ${\mathbb Q}$-algebra homomorphisms
$$
\Omega \otimes {\mathbb Q}\to R.
$$
This is what your book does.