# 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 $a$.