How to calculate localizations?

Question

How to calculate localizations like the following examples? What techniques are used in general in order to calculate localizations?

1) Let $R = K[X,Y]/(XY)$ and $S = \{ X^n \,|\, n \geq 0 \}$. Then $S^{-1}R \cong K[X,X^{-1}]$.

2) Let $R$ be any ring and $t\in R$. Then $R_t:= \{t^n \,|\, n\geq 0 \}^{-1} R \cong R[Y]/(tY-1)$

3) For an interger $a>1$ and a non-zero integer b let $S$ be the multiplicative subset of $\mathbb{Z}/a\mathbb{Z}$ consisting of the residue classes $b^n + a\mathbb{Z}$ for all $n \geq 0$. Then $$S^{-1}(\mathbb{Z}/a\mathbb{Z}) \cong \prod_{p_i \nmid b} \mathbb{Z}/p_i^{l_i}\mathbb{Z}$$ where $a = \prod p_i^{l_i}$ is the prime factorization of $a$ in $\mathbb{Z}$.

Answer 1

Mostly this can be understood well by the universal property of localisation: if $S$ is a multiplicatively closed subset of $R$, and $\phi: R \to Q$ is a homomorphism of rings such that $\phi(s)$ is a unit for all $s \in S$, then $\phi$ factors through $R \to S^{-1}R \to Q$ for some uniquely determined homomorphism $S^{-1}R \to Q$.

The upshot of this is that any relation you can find holding for a general $\phi: R \to Q$ satisfying that property must hold in the localisation $S^{-1}R$. I'll go through the two of the examples above in a bit of depth: after some practise you should be able to just look at these and basically write down the answer.

1. Suppose we have a morphism $\phi: R[x, y]/(xy) \to Q$ where $\phi(x)$ is a unit. Since $\phi(xy) = 0 = \phi(x) \phi(y)$ and $\phi(x)$ is a unit, $\phi(y) = 0$. Now by the universal property of polynomial rings $\phi$ is determined entirely by where it sends $x$, and the only restriction is it must send $x$ to a unit. This property is captured by maps out of the ring $K[x, x^{-1}]$.

2. We want to show that $R[Y]/(tY - 1)$ is the localisation of $R$ at $S=\{1, t, t^2, \ldots\}$, in other words, that it satisfies the universal property above. A homomorphism $\psi: R[Y]/(tY - 1) \to Q$ is the data of a ring homomorphism from $R$, as well as sending $Y$ to an element such that $\psi(Y)\psi(t) = 1$, making $\psi(t)$ a unit. It is not hard to see that this is universal, in the sense that any $\phi: R \to Q$ sending $t$ to a unit factors through here.

As one of the commenters pointed out, the third example is very similar to the first two. Are you able to complete the third one?