In my lecture notes, a cyclic $R$-module is defined as an $R$-module $M$ where there exists some $x \in M$ such that $M=\langle x \rangle_R$. This notation is defined as $$\langle X \rangle_R := \bigcap_{\substack{N\textrm{ an }R\textrm{-submodule, }\\ X \subseteq N}} N.$$
Elsewhere (e.g. http://www.ucl.ac.uk/~ucahjlo/teaching/3201-chapter3-1.pdf) I have seen a cyclic $R$-module defined as any module $M$ which can be written as
$$M = xR := \{xr \mid r \in R\},$$
for some $x \in M.$
Are these definitions equivalent, and if not, which one of them in standard?
I'm probably getting confused here, but I have an example in mind which I believe is 'cyclic' in the first definition, but not the second:
Let the underlying group be $(\mathbb{Z},+,0)$, and the ring $R$ be the field $\{0,1\}$, which $m\cdot 0 = 0$ and $m\cdot 1 = m$ for all $m$.
In the second definition, no matter what we choose for $x$, $xR$ has only two elements ($0$ and $x$). In the first, do we not have $\langle 1 \rangle_R = M$?
Many thanks.
Best Answer
They are the same. Your example does not work, because yours is not a module action, as $$ 0 = m \cdot 0 = m \cdot (1 + 1) = m \cdot 1 + m \cdot 1 = m + m $$ clearly does not hold for $m \ne 0$.
To see the two definitions are the same, note that if a submodule contains $x$, then it contains $x r$ for each $r \in R$, and thus contains $x R$. Since $x R$ is a submodule, it is the smallest submodule containing $x$.