Integral domain and unit of a ring

Lt R be a commutative ring not neccesarily an integral domain. Determine the truth or falsity of the following statements. In each case, give a proof or find a counter-example

Let x, y belongs to R

(i) x.y in R* ---> x in R* , y in R* (R* = R\{0})

(ii) x.y in U(R) ---> x in U(R), y in U(R)

(iii) x.y in R*\ U(R) ---> x in R*\ U(R), y in R*\U(R)

This is my attempt

(i)If x.y in R* then we have x.y not equal to 0 implies that x is nonzero and y is nonzero. So, the statement is true

(ii)I think this statement is false but I dont know how to prove it

(iii)This part I dont know

thank you