Question

Let A ⊆ $\Z$ with 0 ∈ A. For r, s ∈ $\Z$, define
rA = {ra : a ∈ A}, rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group $\Z$ ?

• A. −2A ⊆ A, A + A = A
• B. A = −A, A + 2A = A
• C. A = −A, A + A = A
• D. 2A ⊆ A, A + A = A

Answer 1

Answer: (A), (B), (C)

−2A ⊆ A, A + A = A

Answer 2

The correct option is (A) : −2A ⊆ A, A + A = A, (B) : A = −A, A + 2A = A and (C) : A = −A, A + A = A