Let R and S be rings with identity, : R S a homomorphism of rings

Let R and S be rings with identity, φ : R → S a homomorphism of rings with φ(1_R) = 1_s, and s₁,s₂, ••• , s_n ϵ S such that S_is_j = s_iS_j for all i,j and φ(r)s_i = S_iφ(r) for all r ϵ R and all i. Then there is a unique homomorphism φ̅ : R[x₁, ... , x_n] → S such that φ̅|R = φ and φ̅(x_i) = s_i. This property completely determines R[x₁, ... ,x_n] up to isomorphism.