13. (a) In the ring Z24 nd the sum 18 + 15 and the pro du t 16 · 12. [2 marks℄(b) In the ring Z11 ⊕Z4 nd the sum (6, 2)+(5, 3) and the pro du t (7, 2)(5, 1).[2 marks℄( ) Find the units in the ring Z ⊕ Z and use your answer to determine if thering Z ⊕ Z is a eld.

(a) In the ring Z24, the operations are performed modulo 24.

For the sum 18 + 15: Add the two numbers as usual, 18 + 15 = 33. Then, find the remainder when 33 is divided by 24 (since we are in ring Z24). 33 mod 24 = 9. So, 18 + 15 = 9 in the ring Z24.

For the product 16 * 12: Multiply the two numbers as usual, 16 * 12 = 192. Then, find the remainder when 192 is divided by 24. 192 mod 24 = 0. So, 16 * 12 = 0 in the ring Z24.

(b) In the ring Z11 ⊕Z4, the operations are performed component-wise with the first component modulo 11 and the second modulo 4.

For the sum (6, 2) + (5, 3): Add the first components together and find the remainder when divided by 11: (6 + 5) mod 11 = 0. Add the second components together and find the remainder when divided by 4: (2 + 3) mod 4 = 1. So, (6, 2) + (5, 3) = (0, 1) in the ring Z11 ⊕Z4.

For the product (7, 2) * (5, 1): Multiply the first components together and find the remainder when divided by 11: (7 * 5) mod 11 = 2. Multiply the second components together and find the remainder when divided by 4: (2 * 1) mod 4 = 2. So, (7, 2) * (5, 1) = (2, 2) in the ring Z11 ⊕Z4.

(c) The units in the ring Z ⊕ Z are all pairs (a, b) such that both a and b are units in Z. In Z, the only units are 1 and -1. So, the units in Z ⊕ Z are (1, 1), (1, -1), (-1, 1), and (-1, -1).

A ring is a field if every non-zero element has a multiplicative inverse. In Z ⊕ Z, the element (1, 0) does not have a multiplicative inverse, so Z ⊕ Z is not a field.

(a) In the ring Z24, the operations are performed modulo 24.

For the sum 18 + 15: Add the two numbers as usual, 18 + 15 = 33. Then, take the remainder when 33 is divided by 24. 33 mod 24 = 9. So, 18 + 15 = 9 in Z24.

For the product 16 * 12: Multiply the two numbers as usual, 16 * 12 = 192. Then, take the remainder when 192 is divided by 24. 192 mod 24 = 0. So, 16 * 12 = 0 in Z24.

(b) In the ring Z11 ⊕Z4, the operations are performed component-wise with the first component modulo 11 and the second modulo 4.

For the sum (6, 2) + (5, 3): Add the components separately, (6+5, 2+3) = (11, 5). Then, take the remainder of each component when divided by their respective moduli, (11 mod 11, 5 mod 4) = (0, 1). So, (6, 2) + (5, 3) = (0, 1) in Z11 ⊕Z4.

For the product (7, 2) * (5, 1): Multiply the components separately, (75, 21) = (35, 2). Then, take the remainder of each component when divided by their respective moduli, (35 mod 11, 2 mod 4) = (2, 2). So, (7, 2) * (5, 1) = (2, 2) in Z11 ⊕Z4.

(c) The units in the ring Z ⊕ Z are all pairs (a, b) where a and b are units in Z. In Z, the only units are 1 and -1. So, the units in Z ⊕ Z are (1, 1), (1, -1), (-1, 1), and (-1, -1).

A ring is a field if every non-zero element has a multiplicative inverse. In Z ⊕ Z, the element (1, 0) does not have a multiplicative inverse, so Z ⊕ Z is not a field.