# Definition:Principal Ideal of Ring

## Definition

Let $\left({R, +, \circ}\right)$ be a ring with unity.

Let $a \in R$.

We define:

- $\left({a}\right) = \displaystyle \left\{{\sum_{i \mathop = 1}^n r_i \circ a \circ s_i: n \in \N, r_i, s_i \in R}\right\}$

Then:

- $(1): \quad \forall a \in R: \left({a}\right)$ is an ideal of $R$
- $(2): \quad \forall a \in R: a \in \left({a}\right)$
- $(3): \quad \forall a \in R:$ if $J$ is an ideal of $R$, and $a \in J$, then $\left({a}\right) \subseteq J$. That is, $\left({a}\right)$ is the smallest ideal of $R$ containing $a$.

The ideal $\left({a}\right)$ is called the **principal ideal of $R$ generated by $a$**.

## Also see

- From Principal Ideal is Ideal, $\left({a}\right)$ is a
**principal ideal**if $\left \langle {a} \right \rangle$ is the ideal generated by $a$.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 22$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): $\S 5.21$ - 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): $\S 59$