Definition:Principal Ideal of Ring

Definition
Let $\struct {R, +, \circ}$ be a ring with unity.

Let $a \in R$.

We define:
 * $\ideal a = \displaystyle \set {\sum_{i \mathop = 1}^n r_i \circ a \circ s_i: n \in \N, r_i, s_i \in R}$

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

The ideal $\ideal a$ is called the principal ideal of $R$ generated by $a$.

Also see

 * Principal Ideal is Ideal: $\ideal a$ is a principal ideal if $\gen a$ is the ideal generated by $a$.