Ideal

Definition

Given a Ring $(R,+,\cdot )$, a subset $I$ of $R$ is called an ideal if it satisfies the following conditions:

1. $(I,+)$ is a Subgroup of $(R,+)$.
2. For all $r\in R$ and $x\in I$, $rx\in I$ and $xr\in I$.
   • If only $rx\in I$ holds, $I$ is called a left ideal. If only $xr\in I$ holds, $I$ is called a right ideal. If both conditions hold, $I$ is called a two-sided ideal or simply an ideal.

Characteristics

Ideals have the following characteristics:

• The Opposite Ring of a left ideal is a right ideal, and the same relationship holds for right ideals.
• An ideal is a sub-Pseudoring of the ring $(R,+,\cdot )$.