Principal Ideal of Commutative Ring

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Theorem

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

Let $a \in R$.


Let $Ra$ be the principal left ideal of $R$ generated by $a$.

Let $aR$ be the principal right ideal of $R$ generated by $a$.

Let $\ideal a$ be the principal ideal of $R$ generated by $a$.


Then $Ra = \ideal a = aR$.


Proof

By definition of principal left ideal:

$Ra = \set{r \circ a: r \in R}$

By definition of commutative ring with unity and center of ring:

$a$ is in the center of $R$

From Principal Ideal from Element in Center of Ring:

$\ideal a = R \circ a = \set{r \circ a: r \in R}$

Hence:

$Ra = \ideal a$


We have:

\(\ds Ra\) \(=\) \(\ds \set{r \circ a : r \in R}\) Definition of Principal Left Ideal of Ring
\(\ds \) \(=\) \(\ds \set{a \circ r : r \in R}\) commutativity of ring product $\circ$
\(\ds \) \(=\) \(\ds aR\) Definition of Principal Right Ideal of Ring

$\blacksquare$