Exercise 22, § 2

Let A be a ring (not necessarily commutative) and S a multiplicative subset of A.

(a) Let $\mathscr F$ be the set of left ideals I of A such that, for all $a\in A$, there exists $s\in S$ such that $sa\in I$; this implies in particular that $I\cap S\ne\varnothing$. Show that $\mathscr F$ is topologizing and idempotent.

(b) Let B be a ring and $\phi:A\longrightarrow B$ a ring homomorphism. The ordered pair $(B,\phi)$ is called a ring of left fractions of A with denominators in S if it satisfies the following conditions:

(I) If $\phi(a)=0$, there exists $s\in S$ such that $sa=0$.

(II) If $s\in S$, $\phi(s)$ is invertible in B.

(III) Every element of B is of the form $(\phi(s))^{-1}\phi(a)$, where $a\in A$.

Show that the following properties are equivalent:

$(a)$ The ring A possesses a ring of left fractions with denominators in S.

$(\beta)$ The following conditions hold:

$(\beta_1)$ For all $s\in S$, $a\in A$, there exists $t\in S$ and $b\in A$ such that $ta=bs$.

$(\beta_2)$ If $a\in A$ and $s\in S$ satisfy $as=0$, there exists $t\in S$ such that $ta=0$.

$(\gamma)$ The canonical images of the elements of S in $A_{\mathscr F}$ are invertible.

Moreover these properties imply the following:

$(\delta)$ The principal ideals $As$, where $s\in S$, belong to $\mathscr F$ and every ideal $m\in\mathscr F$ contains one of these principal ideals.

$(\epsilon)$ The left annihilator of every $s\in S$ is contained in $\mathscr F A_{,}$.

$(\zeta)$ For every exact sequence $0\longrightarrow M\xrightarrow{h}N\xrightarrow{p}P\longrightarrow0$ of A-modules, the sequence $0\longrightarrow M_{\mathscr F}\xrightarrow{h_{\mathscr F}}N_{\mathscr F}\xrightarrow{p_{\mathscr F}}P_{\mathscr F}\longrightarrow0$ is exact.

(Show that $(a)\Rightarrow(\beta)\Rightarrow(\gamma)\Rightarrow(a)$. To see that $(\gamma)$ implies $(a)$, note that, for every A-module M and every $x\in\mathscr F M$, there exists $s\in S$ such that $sx=0$. To see that $(\beta)$ implies $(\gamma)$, prove first that $(\beta)$ implies $(\delta)$, $(\epsilon)$ and $(\zeta)$. To show that $(\beta)$ implies $(\zeta)$, establish, using $(\delta)$, that, if $m\in\mathscr F$ and

then there exists s \in S such that As \subset m and v \in \mathrm{Hom}_A(As, N/\mathscr{F}N) such that the diagram

is commutative, i being the canonical injection and $p'$ the mapping derived from $p$ by taking quotients. Consider finally, for all $s \in S$, the exact sequence

where $\mu_s$ is the homothety $\lambda \mapsto \lambda s$; use (E), (\zeta) and Exercise 4 of Algebra, Chapter I, § 8.)

(c) Deduce from (b) that, if A possesses a ring of left fractions (B, $\phi$) with denominators in S, this ring has the following universal property: for every ring homomorphism $f : A \to C$ such that $f(s)$ is invertible in C for all $s \in S$, there exists a unique homomorphism $g : B \to C$ such that $f = g \circ \phi$. In particular B is canonically isomorphic to $A_{\mathscr{F}}$.

(d) The notion of a ring of right fractions of A with denominators in S is similarly defined. Deduce from (c) that, if there exist both a ring of left fractions and a ring of right fractions of A, these two rings are isomorphic.

(e) Let E be a vector space over a field K, $(e_i)_{i \in I}$ a basis of E and T the tensor algebra of E. Let J be a non-empty subset of I distinct from I and S the multiplicative subset of T generated by the $e_i$ such that $i \in J$. Show that T admits no ring of left or right fractions with denominators in S.

(f) With the notation of (e) suppose that I = N and consider the quotient ring A of T by the two-sided ideal generated by the elements $e_0^2$ and $e_0 e_i - e_{i+1} e_0$, for all $i > 0$, and $e_i e_j - e_j e_i$, for $i > 0, j > 0$. Let S' be the multiplicative set generated by the canonical images of the $e_i$ in A, $\epsilon$ or $i > 0$; show that A admits a ring of left fractions, but not a ring of right fractions, with denominators in S'.

(g) Suppose that A admits a ring of left fractions with denominators in S. In the set S x A, let R be the relation between $(s, x)$ and $(t, y)$: "there exists $r \in S$, $u \in A$, $v \in A$ such that $r = us = vt$ and $ux = vy$". Show that R is an equivalence relation and define on the set $B = (S \times A)/R$ a ring structure such that the ordered pair consisting of S and the mapping $\phi : A \to B$, which maps $x \in A$ to the class of $(1, x)$, is a ring of left fractions of A with denominator in S (use the fact that, for $s \in S$ and $t \in S$, there exists $u \in A$ and $v \in A$ such that $r = us = tv$ belongs to S). Consider the case $S = A - \{0\}$ (cf. Algebra, Chapter I, § 9, Exercise 8).