Question

will rate, please show work

3. Problem 3. For each of the following rings, specify the additive identity element, the multiplicative identity element (that is, the unity), and all the unit elements: a) Q: b) ZA;

Answer 1

(a) the additive identity is 0 and multiplicative identity is 1.

The units are all non-zero elements of Q, i.e. Q - {0}

(b) the additive identity is, [0] and the multiplicative identity is, [1]

In, Z₄, we have, [1]•[1] = [1]

&, [3]•[3] = [9] = [1]

So, the units are { [1], [3] }

(c) the additive identity is, [0] & the multiplicative identity is [1]

Since, 7 is a prime, so, the units are all non zero elements of Z₇ , i.e. Z₇ - {[0]}