Thank you. It seems as if Zc =def {nc: n an integer}. But then Zc is not a subring of R. Can I still make sums and products involving Zc?

\(\displaystyle \mathbb{Z}c\) is a subset (not subring) of \(\displaystyle R\) because \(\displaystyle nc \in R\) for all \(\displaystyle n \in \mathbb{Z}.\) so \(\displaystyle rc+nc \in R\), for all \(\displaystyle r \in R, \ n \in \mathbb{Z},\) i.e. \(\displaystyle I=Rc + \mathbb{Z}c \subseteq R.\) (note that we

__cannot__ write \(\displaystyle rc+nc=(r+n)c\) because \(\displaystyle R\) might

not have 1 and so \(\displaystyle n\) might not be in \(\displaystyle R.\)) so you need to show that \(\displaystyle I\) is the smallest ideal of \(\displaystyle R\) which contains \(\displaystyle c\). that is proved very easily: \(\displaystyle I\) is clearly an additive group of \(\displaystyle R\) and if \(\displaystyle s \in R,\) then

\(\displaystyle s(rc+nc)=(sr + ns)c \in Rc \subseteq I.\) also if \(\displaystyle J\) is any ideal of \(\displaystyle R\) which contains \(\displaystyle c,\) then \(\displaystyle Rc \subseteq J\) and \(\displaystyle \mathbb{Z}c \subseteq J\) and thus \(\displaystyle I=Rc + \mathbb{Z}c \subseteq J.\) that means \(\displaystyle I=(c).\)