Let
R be a ring without identity, and let S = R+Z (+ means addition in sets). On the commutative group S, we define a multiplication by

(
r,a).(r´,b) = (rr´ + ar´ + br,ab)
for all (r,a), (r´,b) are in S. Prove that S is a ring with identity.

2. Originally Posted by bogazichili
Let

R be a ring without identity, and let S = R+Z (+ means addition in sets). On the commutative group S, we define a multiplication by

(r,a).(r´,b) = (rr´ + ar´ + br,ab)
for all (r,a), (r´,b) are in S. Prove that S is a ring with identity.
you really ruined the beauty of this problem by putting "urgent" and the unrelated "field" in the title of your post! the idea of embedding a non-unitary ring (as an ideal) in a unitary ring,

although very simple but it's quite nice and helpful. what you need to notice here is that $1_S=(0,1), \ 0_S=(0,0), \ -(r,a)=(-r,-a).$ the rest is just checking the axioms.