The vanishing ideal of the complex torus

I am trying to get my hands on the basics of algebraic geometry, and I am confused about the following.

Let $$(\mathbb{C}^{\times})^n \subset \mathbb{C}^n$$ denote the complex torus. What I know is that to any given algebraic variety $$X \subset \mathbb{C}^n$$, one can associate its vanishing ideal $$I(X) := \lbrace P \in \mathbb{C}[x_1,...,x_n] \ | \ P(x_1,...,x_n) = 0, \ \forall (x_1,...,x_n) \in X \rbrace.$$ Then the ring of regular functions on $$X$$ is defined as the quotient $$\mathbb{C}[x_1,...,x_n] / I(X).$$ For instance, the affine space $$\mathbb{C}^n$$ has vanishing ideal $$\{0\}$$, and therefore its ring of regular functions is the whole ring of polynomials.

For the complex torus, I was told that the ring of regular functions is given by the ring of Laurent polynomials $$\mathbb{C}[x_1,...,x_n,x_1^{-1},...,x_n^{-1}].$$

My questions are the following:

1. How can I get the ring of Laurent polynomials from the above reasoning ?
2. Let I(X) be the vanishing ideal of a subvariety $$X \subset \mathbb{C}^n$$. I have read somewhere that intersecting subvarieties corresponds to adding vanishing ideals. I there a way to interpret the quotient $$\mathbb{C}[x_1,...,x_n,x_1^{-1},...,x_n^{-1}] / I(X)\mathbb{C}[x_1,...,x_n,x_1^{-1},...,x_n^{-1}]$$ in terms of the intersection / union (or something else) of $$X$$ and $$(\mathbb{C}^{\times})^n$$ ? If it helps, one can suppose that $$X$$ is a linear subspace of $$\mathbb{C}^n$$.

Thanks a lot !

The algebraic torus $$(\mathbb{C}^\times)^n$$ is not a closed subvariety of affine $$n$$-space. However, we can see $$(\mathbb{C}^\times)^n$$ as the affine subvariety of $$\mathbb{C}^{2n}$$ with variables $$x_1, x_2, \ldots, x_n, y_1,y_2, \ldots, y_n$$ cut out by the $$n$$ polynomials $$x_iy_i - 1$$. Its affine coordinate ring is then isomorphic to the ring you wrote down.