Ideal

Definition 11. Let \(R\) be a commutative ring and \(\emptyset \ne I \subseteq R\). Then \(I\) is an ideal of \(R\) if:

  1. The sum \(a + b\) is in \(I\) for all \(a, b \in I\).
  2. The product \(ar\) is in \(I\) for all \(a \in I\) and \(r \in R\).

The ideal \(I\) is proper if \(I \ne R\).

Remark 12. Definition 11 says that if \(R\) is a commutative ring, and \(\emptyset \neq I \subseteq R\), then \(I\) is an ideal of \(R\) if:

  1. The set \(I\) is closed under addition.
  2. The set \(I\) is closed under multiplication by any \(r \in R\).

Proposition 13. Let \(I\) be an ideal of a commutative ring \(R\). Then:

  1. The equation \(a \cdot 0 = 0 \cdot a = 0\) holds for all \(a \in R\).
  2. The element \(0\) is in \(I\).
  3. If \(1 \in I\) then \(I\) is not proper.
math definitions
Aug 6, 2023