Q: the number of subring of ring of integer $\mathbb{Z}$ is infinite (T/F)
I know that subring no need exist multiplicative unit in subring. Then for all $n \in \mathbb{N}$, $n\mathbb{Z}$ is subring of $\mathbb{Z}$, the answer above question is T.
But some textbooks says the definition of subring exists multiplicative unit in it(Serge lang, Bourbaki, ...). Then following this definition, the only subring of $\mathbb{Z}$ is $\mathbb{Z}$. therefore the answer above question is F.
Which one is correct definition of subring?
(Above question was given in the teacher certification exam in korea, this exam didn't specify which textbooks in study.)