Let R be a ring without identity. Embed Rina ring S with identity and characteristic zero as in the proof of Identify R with its image in S.

(a) Show that every element of S may be uniquely expressed in the form r1_s + n1_s (r ϵ R, n ϵ Z).

(b) If A is an R-module and a ϵ A, show that there is a unique R-module homomorphism ∫: S → A such that ∫(1_s) = a.