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I am studying algebraic geometry. I looked up the solution of a problem where there is a ring $R[x,\frac{1}{x}]$ ? What are the elements of this ring ?

Is this description correct ?

$$R\left[x,\frac{1}{x}\right]=\left\{\sum_{n=k}^{l}a_nx^n\mathrel{}\middle|\mathrel{}a_n\in R, k\in\mathbb{Z}\right\}$$

(Basically "polynomials" where instead of stopping at the constant term, negative powers are also allowed.)

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    Yes, exactly (with $l\in\mathbb{Z}$ also). – 2017-01-22
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    Also worth noting that this ring is called the ring of Laurent polynomials (in one variable) over $R$. – 2017-01-22
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    We know that $R[x]$ is a PID when $R$ is a field. Can we say something similar about $R[x,\frac{1}{x}]$ ? – 2017-01-22
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    It's a ring of fractions of a P.I.D., hence a P.I.D. itself. – 2017-01-22

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