Yes. In other words, that the object given satisfies the definition of a ring.
Let R and S be commutative rings. Prove that the set of all ordered pairs (r,s) such that r is in R and s is in R can be given a ring structure by defining (r1,s1)+(r2,s2)=(r1+r2,s1+s2) and (r1,s1)*(r2,s2)=(r1*r2,s1*s2).
Let (r1,s1)+(r2,s2)=(r1+r2,s1+s2) and (r1,s1)*(r2,s2)=(r1*r2,s1*s2).
Would I just show we have an abelian group under addition, multiplication is associative and commutative, there is a multiplicative identity elementm and distributive laws hold?