Definition:Principal Ideal of Ring

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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 $a$.


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

  • Results about principal ideals of rings can be found here.