Category:Definitions/Principal Ideals of Rings

This category contains definitions related to Principal Ideals of Rings.
Related results can be found in Category:Principal Ideals of Rings.

Let $\struct {R, +, \circ}$ be a ring with unity.

Let $a \in R$.

We define:

Definition 1

$\ideal a = \ds \set {\sum_{i \mathop = 1}^n r_i \circ a \circ s_i: n \in \N, r_i, s_i \in R}$

Definition 2

$\ideal a$ is the smallest ideal of $\struct {R, +, \circ}$ containing $a$ as an element.

Definition 3

$\ideal a$ is the intersection of all ideals of $\struct {R, +, \circ}$ which contain $a$ as an element.

Definition 4

$\ideal a$ is an ideal of $\struct {R, +, \circ}$ such that every element of $\ideal a$ is of the form $a \circ r$, where $r \in R$

The ideal $\ideal a$ is called the principal ideal of $R$ generated by $a$.