Ideal

Definition m. 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 c. Definition m 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 p. 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