a) Let R b e the ring Z6 and let S = {0, 2, 4} ⊆ R.(i) Use the Subring test to show that S is a subring of R. [4 marks℄(ii) Ea h of R, S has a unity element. State the unity of R and the unityof S.

(i) The Subring Test is a method used to determine whether a subset of a ring is a subring. The test states that a nonempty subset S of a ring R is a subring if for all a, b in S, the difference a - b is in S and the product ab is in S.

Let's apply this test to S = {0, 2, 4} in the ring Z6.

1. Closure under subtraction:

   • 0 - 0 = 0, which is in S.
   • 2 - 0 = 2, 2 - 2 = 0, and 2 - 4 = 4 (since we are working mod 6), all of which are in S.
   • 4 - 0 = 4, 4 - 2 = 2, and 4 - 4 = 0, all of which are in S. So, S is closed under subtraction.

2. Closure under multiplication:

   • 0 * 0 = 0, 0 * 2 = 0, and 0 * 4 = 0, all of which are in S.
   • 2 * 0 = 0, 2 * 2 = 4, and 2 * 4 = 2 (since we are working mod 6), all of which are in S.
   • 4 * 0 = 0, 4 * 2 = 2, and 4 * 4 = 4 (since we are working mod 6), all of which are in S. So, S is closed under multiplication.

Therefore, by the Subring Test, S is a subring of R.

(ii) The unity (or identity) element of a ring is the element that leaves other elements unchanged when multiplied by them.

In the ring Z6, the unity element is 1 because for any element a in Z6, 1 * a = a.

In the subring S = {0, 2, 4}, the unity element is not the same as in Z6 because 1 is not in S. In fact, S does not have a unity element because there is no element in S that leaves other elements unchanged when multiplied by them. For example, 2 * 2 = 4 and 4 * 4 = 4 (mod 6), neither of which are the original elements.