This exercise shows that every ring R can be enlarged (if necessary) to a ring S with unity, having the same characteristic as R. Let S = R x Z if R has characteristic 0, and R x Z_n if R has characteristic n. Let addition in S be the usual addition by components, and let multiplication be defined by (r₁, n₁)(r₂, n₂) = (r₁r₂ + n₁ · r₂ + n₂ · r₁, n₁n₂) where n · r has the meaning explained in Section 18. 

a. Show that S is a ring. 

b. Show that S has unity. 

c. Show that S and R have the same characteristic. 

d. Show that the map ∅ : R → S given by ∅(r) = (r, 0) for r ∈ R maps R isomorphically onto a subring of S.