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 |
$\blacksquare$