I confused about how to define the multiplication for the ring of Witt vectors $(W(R)=1+tR[[t]],\cdot,*)$, for every commutative ring. The addition '$\cdot$' is naturally defined by the multiplication of polynomials in $R[[t]]$. However for the multiplication '$*$', in C. Weibel's 'The K-book', he says it is sufficiently just define $$(1+rt)*f(t)=f(rt),\text{ for any } r\in R, f\in W(R),$$ then we could automatically get the general definition $f(t)*g(t),f,g\in W(R)$. However I could not deduced the multiplication $(1-r_1t^m)*(1-r_2t^n)$ by the upside formula and the law of association.
When the commutative ring $R$ is a field (especially algebraic closed field), this would not be a problem, since we always have decomposition $f(t)=(1-r_1t)\cdots (1-r_nt)\cdots$ for every $f(t)$. But for the general case, when $R$ is an arbitrary commutative ring, I do not know how to compute multiplications.
This problem have been confusing me for a long time, since The multiplication '$*$' of the ring $W(K)=1+tK[[t]]$ in C.Weibel's 'The K-book'. I would be very grateful if someone could provide a useful answer.