Let I be a subset of R, where R is a commutative ring and let I be an ideal of R.

Show that:

(i) I = R <=> 1 belongs to I <=> R/I = 0

(ii) For a in R, (a) = 0 ={0} <=> a = 0

and (a) = R <=> a in U(R)

Where (a) = Ra = {ra: r in R} is the principal ideal generated by a

This is my attempt:

(i) (=>) Since I = R and 1 is in R, so 1 belongs to I

(=>) 1 in R, then 1 + I = 0+I = R/I = 0

Hence R/I = 0 = { 0+I}

Coversely, (<=) R/I = 0, so 1+ I = 0 + I implies 1 belongs to I and since I in R then I = R

Is that enough for this argument?

Part(ii) I am not sure how to do it, please give some hints

Thank you so much in advanced