MAT3702 Assignment TWO 2024

Question 1

𝑍6 = {0, 1, 2, 3, 4, 5}


𝐿 = {0, 2, 4}

(i)

𝐿 has three elements which are also elements of 𝑍6. So, 𝐿 is a nonempty subset of 𝑍6


(ii)

0 + (− 0) = 0 ∈ 𝐿

2 + (− 2) = 0 ∈ 𝐿

4 + (− 4) = 0 ∈ 𝐿

0 + (− 2) = 4 ∈ 𝐿

0 + (− 4) = 2 ∈ 𝐿

2 + (− 0) = 2 ∈ 𝐿

2 + (− 4) = 4 ∈ 𝐿

4 + (− 0) = 4 ∈ 𝐿

4 + (− 2) = 2 ∈ 𝐿

So, for all 𝑥, 𝑦 ∈ 𝐿, 𝑥 + (− 𝑦) ∈ 𝐿

(iii)

0×0=0∈𝐿

2×2=4∈𝐿

4 × 4 = 16 = 4 ∈ 𝐿

0×2=0∈𝐿

0×4=0∈𝐿

, 2×0=0∈𝐿

2×4=8=2∈𝐿

4×0=0∈𝐿

4×2=8=2∈𝐿

So, for all 𝑥, 𝑦 ∈ 𝐿, 𝑥 × 𝑦 ∈ 𝐿

By i, ii and iii, 𝐿 is a subring of 𝑍6


The identity element of 𝑍6 is 1, because 1 × 𝑥 = 𝑥 = 𝑥 × 1 for all 𝑥 ∈ 𝑍6


But 1 ∉ 𝐿. So, 𝐿 does not contain the identity element.


Question 2

𝑅 = {0, 2, 4, 6, 8} modulo 10

The identity element for addition is 0 because 0 + 𝑥 = 𝑥 = 𝑥 + 0 for all 𝑥 ∈ 𝑅

Multiplication table for 𝑅



× 0 2 4 6 8

0 0 0 0 0 0

2 0 4 8 12 = 2 16 = 6

4 0 8 16 = 6 24 = 4 32 = 2

6 0 12 = 2 24 = 4 36 = 6 48 = 8

8 0 16 = 6 32 = 2 48 = 8 64 = 4



The row and column with 6 both have the elements of 𝑅 in the order 0, 2, 4, 6, 8

The identity element for multiplication is 6