Definition:Principal Ideal

Definition

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

Let $a \in R$.

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

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

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 22$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 5.21$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 59$