Principal Ideal from Element in Center of Ring

Theorem
Let $$\left({R, +, \circ}\right)$$ be a ring with unity whose zero is $$0_R$$ and whose unity is $$1_R$$.

Let $$b \in R$$ be in the center of $$R$$.

Then:
 * $$\left({b}\right) = R \circ b = \left\{{x \circ b: x \in R}\right\}$$

where $$\left({b}\right)$$ is the principal ideal generated by $b$.

Proof
Let $$J = R \circ b$$

The center of $$R$$ is defined as:
 * $$Z \left({R}\right) = \left\{{x \in R: \forall s \in R: s \circ x = x \circ s}\right\}$$

Therefore:
 * $$R \circ b = b \circ R = \left\{{x \circ b: x \in R}\right\} = \left\{{b \circ x: x \in R}\right\}$$

and so:
 * $$x \in J \implies x \circ b \in J \and b \circ x \in J$$

Thus $$J$$ is an ideal of $$R$$ and so $$J = \left({b}\right)$$.