Zero Element Generates Null Ideal

Theorem

Let $(R,+,\circ)$ be a ring whose zero is $0_R$.

For $r \in R$, let $(r)$ denote the ideal generated by $r$.

Then $(0_R)$ is the null ideal.

Proof

By definition,

$(0_R) = \{ r \circ 0_R : r \in R \}$

but for each $r \in R$ we have by Ring Product with Zero that $r \circ 0_R = 0_R$ for all $r \in R$.

Therefore $(0_R)$ is trivial.

$\blacksquare$