# Definition of principal ideal in rings [closed]

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

## closed as off-topic by GNUSupporter 8964民主女神 地下教會, callculus, José Carlos Santos, Leucippus, YuiTo ChengMay 5 at 1:09

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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.
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 $$R$$.