Meaning of radical ideal. radical ideal is nilpotent set on quotient ring.

1. Meaning of radical ideal
Hanpen Robot
(2015/11/19)

2. Definition of radical ideal 𝐼
𝐼 = 𝑥 ∈ 𝑅 ∃𝑛 ∈ ℕ+
, 𝑥 𝑛
∈ 𝐼 }
when 𝐼 is an ideal of ring 𝑅.

3. Meaning of radical ideal 𝐼
𝑥 ∈ 𝐼 ⇔ ∃𝑛 ∈ ℕ+
, 𝑥 𝑛
∈ 𝐼
⇒ ∃𝑛 ∈ ℕ+
, 𝑥 + 𝐼 𝑛
= 0
When 𝑥 + 𝐼 ∈ 𝑅/𝐼 , 𝑅/𝐼 is quotient ring.

4. Meaning of radical ideal 𝐼
∴ 𝑥 ∈ 𝐼 ⇒ ∃𝑛 ∈ ℕ+
, 𝑥 + 𝐼 𝑛
= 0
if 𝑥 ∈ 𝐼
then 𝑥 + 𝐼 becomes nilpotent on 𝑅/𝐼.

5. THE END