general localization of a commutative ring

We define a localization of a commutative ring in a more general situation than in subsection 7.3.

DEFINITION 07.37   Let $A$ be a commutative ring. Let $S$ be its subset. We say that $S$ is multiplicative if

1. $1\in S$
2. $x,y \in S \ \implies \ x y \in S$

holds.

DEFINITION 07.38   Let $S$ be a multiplicative subset of a commutative ring $A$. Then we define $A[S^{-1}]$ as

$$A[\{X_s ; s \in S\}]/(\{ s X_s -1; s \in S\})$$

where in the above notation $X_s$ is a indeterminate prepared for each element $s \in S$.) We denote by $\iota_S$ a canonical map $A\to A[S^{-1}]$.

LEMMA 07.39   Let $S$ be a multiplicative subset of a commutative ring $A$. Then the ring $B=A[S^{-1}]$ is characterized by the following property:

Let $C$ be a ring, $\varphi:A\to C$ be a ring homomorphism such that $\varphi(s)$ is invertible in $C$ for any $s \in S$. Then there exists a unique ring homomorphism $\psi=\phi[S^{-1}]:B\to C$ such that

$$\varphi=\psi \circ \iota_S$$

holds.

COROLLARY 07.40   Let $S$ be a multiplicative subset of a commutative ring $A$. Let $I$ be an ideal of $A$ given by

$$I=\{ x \in I; \exists s \in S$$    such that $$s x=0\}$$

Then (1) $I$ is an ideal of $A$. Let us put $\bar{A}=A/I$, $\pi:A\to \bar{A}$ the canonical projection. Then

(2) $\bar{S}=\pi(S)$ is multiplicatively closed.

(3) We have

$$A[S^{-1}]\cong\bar{A}[\bar{S}^{-1}]$$

(4) $\iota_{\bar{S}}: \bar{A}\to \bar{A}[\bar{S}^{-1}]$ is injective.

EXAMPLE 07.41   $A_f=A[S^{-1}]$ for $S=\{1,f,f^2,f^3,f^4,\dots\}$. The total ring of quotients $Q(A)$ is defined as $A[S^{-1}]$ for

$$S=\{ x \in A; x$$    is not a zero divisor of A$$\}.$$

When $A$ is an integral domain, then $Q(A)$ is the field of quotients of $A$.

DEFINITION 07.42   Let $A$ be a commutative ring. Let $\mathfrak{p}$ be its prime ideal. Then we define the localization of $A$ with respect to $\mathfrak{p}$ by

$$A_\mathfrak{p}=A[ (A\setminus \mathfrak{p})^{-1}]$$