I have question regarding the localization for ring. I am just started reading from Reid, Undergraduate Commutative Algebra. I got stuck in example 1 page 85. This is the example.
Let $ A=k[x,y]/(xy)$ be a ring and $ S=\{1,x,x^2,.......\}$. Then $$S^{-1}A=k[X,X^{-1}].$$
Note that $f(A)=k[X]\subset K[X,X^{-1}]$; my question I want to know how $S^{-1}A=k[X,X^{-1}]$; and $$f(A)=k[X]\subset K[X,X^{-1}].$$ Maybe it is not that difficult it is just my first example. Any idea will be helpful?