Definition:Principal Ideal of Ring/Definition 4
Definition
Let $\struct {R, +, \circ}$ be a ring with unity.
Let $a \in R$.
We define:
- $\ideal a$ 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
- Results about principal ideals of rings can be found here.
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 59$. Principal ideals in a commutative ring with a one
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ideal