What is the meaning of $\mathbb C[t]_t$ in the theory of local ring? 
I was reading local ring, multiplicative closed set, localization etc..where it was written that $(f(t)/t^m) \in \mathbb C[t]_t$ for some $m \in \mathbb N$ and $f(t)\in \mathbb C[t].$ Here what is the meaning of $\mathbb C[t]_t$ ?

Please someone help..
Thank you..
 A: The four typical meanings I would see that notation given are:


*

*The completion of $\mathbb{C}[t]$ at the prime ideal $(t)$ — that is, the power series ring $\mathbb{C}[[t]]$.

*The local ring of $\mathbb{C}[t]$ at the prime ideal $(t)$ — that is, the ring of all rational functions whose denominator is not divisible by $t$.

*The ring obtained by inverting $t$ — that is, $\mathbb{C}[t, t^{-1}]$. It is the ring of all polynomials in $t$ and $t^{-1}$, or equivalently the ring of all rational functions whose denominator is a power of $t$.

*The quotient ring $\mathbb{C}[t] / (t)$.


The last is, I think, by far the least common. The third is the next least common  — but given the context you saw it in, it's the only one that makes sense.
Since you are talking about the basics, note:


*

*The second bullet above is obtained by inverting the multiplicatively closed subset of all polynomials that aren't in the ideal $(t)$

*The third bullet above is obtained by inverting the multiplicatively closed set generated by $t$ — that is, the set of powers of $t$.

