# Definition:Principal Ideal of Ring

## Definition

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

### 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 $a$**.

## Notation

From Principal Ideal of Commutative Ring the notions of principal left ideal, principal right ideal and **principal ideal** coincide.

So often, in some sources, a **principal ideal** of a commutative ring with unity is denoted as $a R$.

This is done most often in the case where it is important to identify the ring that the **principal ideal** belongs to.

The notation $a R$ is often used when the ring $R$ in question is the integers $\Z$ or the $p$-adic integers $\Z_p$.

So it is common for $n \Z$ to denote the **principal ideal of $\Z$ generated by $n$** and $p^k\Z_p$ to denote the **principal ideal of $\Z_p$ generated by $p^k$**.

## Also see

- Principal Ideal is Ideal: $\ideal a$ is a
**principal ideal**if $\gen a$ is the ideal generated by $a$.

- Results about
**principal ideals of rings**can be found**here**.