Let (S, +, ) and (T, +', ') be two rings. For R = S T,

Question:

Let (S, +, •) and (T, +', •') be two rings. For R = S × T, define addition "⊕" and multiplication "⊙" by
(s1, t1) ⊕ (s2, t2) = (S1 + S2, t1 +' t2),
(s1 t1) ⊙ (s2, t2) = (S1 • s2, t1. t2).
(a) Prove that under these closed binary operations, R is a ring.
(b) If both S and T are commutative, prove that R is commutative.
(c) If S has unity us and T has unity uT, what is the unity of R?
d) If S and T are fields, is R also a field?