Exercise 14, § 9

Let K be an ordered commutative field and G a group. Prove that the algebra of the group G over K does not admit a nonzero (left or right) nil ideal. (For $x=\sum a_ge_g$, set $t(x) =a_1$ and $x^*=\sum a_ge_{g^{-1}}$. Observe that if $x$ is nonzero, then we have $t((xx^*)^{2k})>0$ for every $k\geqslant 0.$) Prove the same property for the algebra of G over the field $K(i)$, where $i^2=-1$.