Q3 Let S = R² with addition and multiplication in S defined by

(a, b) + (x, y) = (a + x, b+y),

(a, b) x (x, y) = (ax, by).

Throughout this question you may assume that R is a field.

(i) Show that (x, y) is an additive identity element in S if and only if (x, y) = (0,0).

(ii) Show that (x, y) is a multiplicative identity element in S if and only if

(x, y) = (1, 1).

(iii) Verify that the distributive law holds in S.

(iv) By considering the element (1,0), or otherwise, show that S is not a

field.