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

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- Nov 14th 2011, 12:58 PMThatPinkSockAbstract algebra: commutative ring proof
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** - Nov 14th 2011, 01:00 PMTinybossRe: Abstract algebra: commutative ring proof
What do you normally check to see whether a subset of a ring is an ideal?

- Nov 14th 2011, 05:03 PMThatPinkSockRe: Abstract algebra: commutative ring proof
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.