Definition of principal ideal in rings

Question

Can an improper ideal ($\varnothing$ or $R$) be a principal one in the ring $R$?

Answer 1

Proper or not, an ideal is an additive subgroup of $R$, therefore it isn't empty. $\{0\}$ is a principal ideal, its generator being $0$. $R$ is principal in rings with $1$, and $1$ itself is its most notable generator; it may not be principal in rings without unity.

Answer 2

The ideal $\langle\{1\}\rangle = R$ is principle, where $1$ is the unit element in $R$, and the ideal $\langle\emptyset\rangle = \{0\}$ is principle, where $0$ is the zero element in $R$.