Herstein Abstract Algebra. Ch 4. Sect 3. Problem 1.
If R is a commutative ring and e ∈ R, let L(a) = {x ∈ R | xa = 0}. Prove that L(a) is an ideal of R
0 = (0)a ∈ L(a),
so L(a) is non-empty.
x₁,x₂ ∈ L(a)
⇒ x₁a = x₂a = 0
⇒ (x₁ + x₂)a = x₁a + x₂a = 0,
so L(a) is closed under addition.
r ∈ R, x ∈ L(a)
⇒ (rx)a = r(xa) = r(0) = 0
⇒ rx ∈ L(a),
so L(a) is closed under multiplication by elements of R.
→ L(a) is an ideal of R.