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.