Definition 3

Let A be a ring, S a subset of A, $\overline{S}$ the multiplicative subset of A generated by S and M an A-module. Then the module of fractions of M defined by S and denoted by $M[S^{-1}]$ or $\overline{S}^{-1}M$ is the $A[S^{-1}]$-module $M \otimes_A A[S^{-1}]$.

In this chapter we shall usually denote by $i_M^S$ the canonical mapping $m \mapsto m \otimes 1$ of M to

Remarks (1) Clearly $M[\overline{S}^{-1}] = M[S^{-1}]$. (2) For $m \in M$ and $s \in \overline{S}$ we also write $m/s$ for the element $m \otimes (1/s)$ of $M[S^{-1}]$. Every element of $M[S^{-1}]$ is of the form, for such an element is of the form $\sum_i m_i \otimes (a_i/s)$, where $m_i \in M, a_i \in A, s \in S$ (no. 1, Remark 2), and

hence $\sum_i m_i \otimes (a_i/s) = m \otimes (1/s)$, where $m = \sum_i a_i m_i$. Then

(4)

(5)

where $m, m' \in M, a \in A$ and $s, s' \in S$.

(3) If S is the complement of a prime ideal p of A, we write M_p instead of S^{-1}M.

(4) Let M be an A[S^{-1}]-module; if M is considered canonically as an A-module, i_M^S is a bijection, for the ordered pair consisting of M and the identity mapping 1, is also trivially a solution of the universal mapping problem solved by M[S^{-1}] and i_M^S. Then M is identified with M[S^{-1}].