4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a multiplication as follows: (A, z) + (µ, w) := (A+µ,z +w) and (λ, 2) . (μ, w) :(μ, λυ + μ). Show that: (i) R with these two operations is a commutative ring; (ii) R, {(0,0)}, and I:= { (0, 2) | z € R} (0, z) are the only ideals in R; and (iii) I = Nil(R).

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