\magnification=1200 \nopagenumbers \baselineskip=13pt
\parskip=4pt plus 1pt
\topskip=35pt \hsize 15truecm \vsize 21.25truecm
\def\itemitemitem{\par\indent\indent\hangindent=3\parindent\textindent}
\def\itemitempegao{\hangindent=2\parindent\textindent}
\def\itemitemitempegao{\hangindent=3\parindent\textindent}
\font\ver=cmcsc10 \font\bmath=cmmib10

%%%%%%%%%%%

\font\tenmsb=msbm10
%\font\eightmsb=msbm8
\font\sevenmsb=msbm7
%\font\sixmsb=msbm6
\font\fivemsb=msbm5
\newfam\msbfam
\textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb
\scriptscriptfont\msbfam=\fivemsb
\def\bbb{\fam\msbfam\tenmsb}% math symbols b, family 11

%%Blackboardbold
\def\AA{{\bbb A}}
\def\BB{{\bbb B}}
\def\CC{{\bbb C}}
\def\DD{{\bbb D}}
\def\EQ{{\bbb EQ}}
\def\HH{{\bbb H}}
\def\II{{\bbb I}}
\def\JJ{{\bbb J}}
\def\KK{{\bbb K}}
\def\MM{{\bbb M}}
\def\NN{{\bbb N}}
\def\OO{{\bbb O}}
\def\OQ{{\bbb OQ}}
\def\QQ{{\bbb Q}}
\def\RR{{\bbb R}}
\def\VV{{\bbb V}}
\def\ZZ{{\bbb Z}}

%%%%%%%%%%%%%

\font\teneufm=eufm10
%\font\eighteufm=eufm8
\font\seveneufm=eufm7
%\font \sixeufm=eufm6
\font\fiveeufm=eufm5
\newfam\eufmfam
\textfont\eufmfam=\teneufm
%\scriptfont\eufmfam=\seveneufm
%\scriptscriptfont\eufmfam=\fiveeufm
\def\fr{\fam\eufmfam\teneufm}% fraktur, family 8


\def\Gl{{\fr gl}}
\def\Ff{{\fr F}}
\def\Fsl{{\fr fsl}}
\def\Fsp{{\fr fsp}}
\def\Fo{{\fr fo}}
\def\Oo{{\fr o}}
\def\Sp{{\fr sp}}

%%%%%%%%%%%%%automatic numbering device%%%%%%%%%%%%%
\newcount\parracount
\newcount\recount
\parracount=0
\recount=0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\re#1#2{\advance\recount by 1
\edef#1{\number\parracount .\number\recount}
\def\nnnn{#2}
\ifx\nnnn\empty\vskip 3 pt plus 1 pt{\bf #1 }\else \vskip 3 pt
plus 1 pt{\bf #1}{ \ver #2. }\it\fi}
\def\endre{\par\rm}
\def\Proof{{\it Proof. }}
\def\endproof{~\vrule height 1.2ex width 1.2ex depth -0.06ex \par}
%\def\endproof{}

\font\parrafont=cmb10
\def\parrafo#1{\vskip-\lastskip\vskip 20 pt plus 10 pt\advance\parracount by
1\recount=0  {\bf \number\parracount.\ \ignorespaces {  #1.}}\par\nobreak\vskip 3 pt plus 3 pt\nobreak}


\def\ad{\mathop{\rm ad}\nolimits}
\def\Alt{\mathop{\rm Alt}\nolimits}
\def\Ann{\mathop{\rm Ann}\nolimits}
\def\Der{\mathop{\rm Der}\nolimits}
\def\dim{\mathop{\rm dim}\nolimits}
\def\End{\mathop{\rm End}\nolimits}
\def\Extr{\mathop{\rm Extr}\nolimits}
\def\Evann{\mathop{\rm Evann}\nolimits}
\def\Heart{\mathop{\rm Heart}\nolimits}
\def\Hom{\mathop{\rm Hom}\nolimits}
\def\IDer{\mathop{\rm IDer}\nolimits}
\def\Id{\mathop{\rm Id}\nolimits}
\def\id{\mathop{\rm id}\nolimits}
\def\Im{\mathop{\rm Im}\nolimits}
\def\Ker{\mathop{\rm Ker}\nolimits}
\def\Mat{\mathop{\rm Mat}\nolimits}
\def\PDer{\mathop{\rm PDer}\nolimits}
\def\span{\mathop{\rm span}\nolimits}
\def\dim{\mathop{\rm dim}\nolimits}
\def\Soc{\mathop{\rm Soc}\nolimits}
\def\TKK{\mathop{\rm TKK}\nolimits}
\def\char{\mathop{\rm char}\nolimits}

\def\directsum{\hbox{\hbox to
2pt{\hss}\hbox{\vbox{\hrule\hbox{\vrule$+$\vrule}\hrule}}\hbox to
2pt{\hss}}}
\def\R#1{\relax\ifmmode R^{(#1)}\else $R^{(#1)}$\fi}

\def\Ii{{\cal I}}

%%%%%%%%%%%%%% FUENTES %%%%%%%%%%%%%%%%
\font\rms=cmr8 \font\bfs=cmbx8 \font\its=cmti8 \font\maths=cmmi8
\font\tit=cmb10 scaled 1300

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\headimpar{ {\ver } \hss {\rm \folio} }
\def\headpar{{\rm \folio}\hfill
{\ver } \hss }

\headline={\ifnum\pageno=1{\hfil {\rm\folio}}\else
\ifodd\pageno\headimpar\else\headpar\fi\fi}
%\footline={\ifnum\pageno=1\hfil\tenrm{\rm
%\folio}\hfil\else\hfil\fi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\def\ad{\mathop{\rm ad}\nolimits}
\def\Alt{\mathop{\rm Alt}\nolimits}
\def\Ann{\mathop{\rm Ann}\nolimits}
\def\Der{\mathop{\rm Der}\nolimits}
\def\dim{\mathop{\rm dim}\nolimits}
\def\End{\mathop{\rm End}\nolimits}
\def\Extr{\mathop{\rm Extr}\nolimits}
\def\Evann{\mathop{\rm Evann}\nolimits}
\def\Heart{\mathop{\rm Heart}\nolimits}
\def\Hom{\mathop{\rm Hom}\nolimits}
\def\IDer{\mathop{\rm IDer}\nolimits}
\def\Id{\mathop{\rm Id}\nolimits}
\def\Im{\mathop{\rm Im}\nolimits}
\def\Ker{\mathop{\rm Ker}\nolimits}
\def\Mat{\mathop{\rm Mat}\nolimits}
\def\span{\mathop{\rm span}\nolimits}
\def\TKK{\mathop{\rm TKK}\nolimits}
\def\directsum{\hbox{\hbox to
2pt{\hss}\hbox{\vbox{\hrule\hbox{\vrule$+$\vrule}\hrule}}\hbox to
2pt{\hss}}}
\def\R#1{\relax\ifmmode R^{(#1)}\else $R^{(#1)}$\fi}

\def\Ii{{\cal I}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%    Encabezados   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\headimpar{\hfil\ver Construction of  $Q_{max}^l(R)$ \hfil{\rm
\folio}}
\def\headpar{{\rm \folio}\hfil\ver   Artacho C\'{a}rdenas, G\'{o}mez Lozano and Ruiz Calvi\~{n}o  \hfil}
\headline={\ifnum\pageno=1\hfil\else\ifodd\pageno\headimpar\else\headpar\fi\fi}
\footline={\ifnum\pageno=1\hfil\tenrm{\rm
\folio}\hfil\else\hfil\fi}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%   Referencias   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\AMI{1}
\def\AGR{2}
\def\BeMik{3}
\def\EKI{4}
\def\FGI{5}

\def\GSalt{6}
\def\Lam{7}

\def\U{8}
\def\ZSSS{9}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5
\newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\bbb{\fam\msbfam\tenmsb}
% math symbols b, family 11 \font\tit=cmb10 scaled 1400 \font\ver=cmcsc10

%%Blackboardbold

\def\CC{{\bbb C}}
\def\NN{{\bbb N}}
\def\RR{{\bbb R}}
\def\ZZ{{\bbb Z}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%   ART\'{I}CULO   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\null \vskip 0.1 truecm \centerline{\ver CONSTRUCTING  THE  MAXIMAL LEFT QUOTIENT RING} \centerline{\ver  OF AN ALTERNATIVE RING.}





\vskip 1truecm \centerline{\ver Laura Artacho C\'{a}rdenas, Miguel G\'{o}mez Lozano\footnote{$^1$}{\rms Partially supported by the MCYT and the Fondos FEDER,
BFM2001-1938-C02-01, by the ``Plan Andaluz de Investigaci\'on y desarrollo Tecnol\'ogico", FQM-264, and proyecto de cooperaci\'{o}n con Marruecos de la
UCA.}} \centerline {And} \centerline{\ver Jorge Ruiz Calvi\~{n}o.}

\vskip.5 truecm \centerline{\it Departamento de \'Algebra Geometr\'\i a y Topolog\'\i a, Universidad de M\'alaga,} \centerline{\it Campus de
Teatinos, 29071 M\'alaga, Spain.} \vskip.5truecm \centerline{\it e-mail addresses: lartacho@agt.cie.uma.es, magomez@agt.cie.uma.es} \centerline{\it
and jruiz@agt.cie.uma.es} \footnote{}{\rms 2000 Mathematical Subject Classification: 16S90, 17D05.} \vskip 1truecm


\medskip

\centerline{  ABSTRACT} \medskip
 We study the notion of general left quotient ring of an
alternative ring and show the existence of a maximal left quotient ring for every alternative ring that is a left quotient ring of itself.



  \parrafo{Introduction}


The theory of rings of quotients has its origins between  1930 and 1940, in the works of O. Ore and  K. Osano on the construction of the total ring
of fractions. In that decade, Ore gave a necessary and sufficient condition for a ring $R$ to have a (left) classical ring of quotients (left Ore
condition). At the end of the 50's, Goldie, Lesieur and Croisot characterized the associative rings that are classical left orders in semiprime and
left artinian rings [\Lam, Chapter IV] (result known as Goldie's Theorem).


Later on in 1956, Y. Utumi introduced the notion of a general ring of left quotients [\U] and proved that the rings without right zero divisors  are
precisely those which  have  maximal left quotient rings.


Following Goldie's idea of characterizing certain types of rings via a suitable envelope, R.E. Johnson characterized those rings $R$ whose maximal
left quotient rings are von Neumann regular, see [\Lam, (13.36)], and P. Gabriel specialized it further by giving characterizations for those rings
whose maximal left quotient rings are semisimple, i.e., isomorphic to a finite direct product of rings of the form $End_{\Delta}(V)$ for suitable
finite dimensional  left vector spaces $V$  over a division ring $\Delta$, see [\Lam, (13.40)].

It is natural to ask whether similar notions and results can be obtained for alternative rings.


In [\EKI] H. Essannouni and A. Kaidi  gave a version for alternative rings of the classical Goldie theory and in [\GSalt] the authors, dealing  with
orders in no necessarily unital alternative rings (see [\AMI] and [\FGI] for associative rings) extended to alternative rings the results and notions
by Fountain and Gould, and \'Ahn and M\'arki.

In 1989, K.L. Beidar and A.V. Mikhalev, interested in the structure of nondegenerate and purely alternative algebras, introduced  what they referred
to as the almost classical localization of an algebra and described, using the theory of orthogonally complete algebraic systems, the structure of
this type of algebras, see [\BeMik, Theorem 2.12]. This construction, that only works when the  associative center coincides with the center (which
is a property of nondegenerate and purely alternative rings), coincides with our maximal left quotient ring in these particular conditions,  see
(2.16)(5).

 The notion of left quotient ring for alternative rings was introduced in [\GSalt] in order to  relate, in the most general way,
 properties between a ring
 and its rings of quotients.  We want to point out that, following [\EKI], ``denominators" were taken in the associative center of the alternative ring.

  In this paper we give the ideas that  gave rise to  the construction of the  maximal left quotient ring of an alternative ring $R$, see [\AGR],
   in the most general case, i.e., when $R$ is a left quotient ring of
itself, equivalently when $R$ is a faithful $N(R)$-module. In general, the  maximal left quotient ring is a non-associative ring whose
 associator  is an alternating function of its arguments, see Theorem 3.8.
  Furthermore, it is an alternative ring when   $D(R)$ is semiprime or 2-torsion free.


\parrafo{Preliminaries}
\medskip

\re{\defI}{} The following three basic central subsets can be considered in an alternative ring $R$: the {\it associative center} $N(R)$, the {\it
commutative center} $K(R)$, and the {\it center} $Z=Z(R)$, defined by:
                      $$\eqalign{ N(R)&=\{ x \in R \ \vert  \ (x, R, R) = (R, x, R) = (R, R, x) = 0 \}, \cr
                    K(R)&=\{ x \in R \ \vert  \ [ x, R]=0\}, \cr
                     Z(R) &= N(R) \cap K(R),\cr} $$
where $ \lbrack x, y \rbrack = xy-yx$ denotes the {\it commutator} of two elements $ x, y \in R $ and $ (x, y, z) = (xy)z - x(yz) $ is the {\it
associator} of  three elements $ x, y, z $ in $ R $.
\endre


\re{\defII}{} The defining axioms for an {\it alternative} ring
$R$ are the {\it left} and the {\it right alternative laws}:
                                   $$ (x, x, y)=0=(y, x, x) $$
  for every $x,y\in R$. As a consequence
we have the fact that the associator is an alternating function of
its arguments. The standard reference for alternative rings is
[\ZSSS].
\endre

\re{\defIII}{} The  associative nucleus and the associator ideal
of an alternative ring will be very important notions in this
theory. Given a ring $ R $,  every ideal contained in the
associative center of $R$ will be called {\it nuclear ideal}. The
largest nuclear ideal of $R$ will be the {\it associative
nucleus},  denoted by $U(R)$.  By $D(R)$ we will mean the {\it
associator ideal}, i.e., the ideal of $R$ generated by the set
$(R, R, R)$ of all associators.
\endre

\re{\defIIII}{} From now on, for an alternative ring $R$, $R^1$ will denote its unitization, that is, $R$ if the ring is unital, or $R \times {\bbb
Z}$ with product $(x, m) (y, n) := (xy + nx + my, mn)$ if $R$ has no unity.\endre


\re{\defIV}{} An alternative ring without nonzero {\it trivial ideals}  (i.e., ideals with zero multiplication) is called  {\it semiprime}. By
[\ZSSS, Exercise 9.1.8], every semiprime alternative ring does not contain nonzero trivial left (right) ideals. An element $a$ of an alternative ring
$R$ is called an {\it absolute zero divisor} if $aRa = \{0\}$. An alternative ring $R$ is called {\it nondegenerate} (or {\it strongly semiprime}) if
$R$ does not contain nonzero absolute zero divisors.
\endre

%\re{\defV}{}  We recall that for every  nonempty subset $X$ of an
%alternative ring $R$,  the {\it left annihilator} of $X$ is
%defined as the set $lan(X) =\{a \in R \  \vert \  ax=0 \  \hbox{ for all} \  x \in X\},$ written  $lan_R(X)$ when it is necessary to emphasize the
%dependence on $R$.  Similarly the {\it right annihilator} of $X$, $ran(X)=ran_R(X)$, is defined by $ ran(X)=\{a \in R \  \vert \ xa=0 \  \hbox{ for
%all} \  x \in X\}.$ We also write $ann(X)= ann_R(X):=$ $lan(X) \cap ran(X)$ to denote the {\it annihilator} of  $X$. In general the left (right)
%annihilator of a subset $X$ of an alternative ring $R$ does not have to be a left (right) ideal, however, it is true if $X$ is a right (left) ideal
%of $R$ or if $X\subset N(R)$.
%\endre


\re{\defi}{} The notion of left quotient ring in the setting of alternative rings was introduced in [\GSalt], where the relationship among classical,
Fountain and Gould and this type of rings of quotients was established.

 Let $R$ be a subring of an alternative ring $Q$. We recall that $Q$ is a {\it left quotient ring}
of $R$, denoted  by $R\le_q Q$, if:

\item{(1)} $N(R)\subset N(Q)$ and
\item{(2)} for every $p,q\in Q$, with $p\ne 0$, there exists $r\in N(R)$ such that $rp\ne 0$ and $rq\in R$.
\endre

Note that $R$ and $Q$ can be seen as left $N(R)$-modules and that
condition $(2)$ of the  previous definition means that $R$ is a
dense left $N(R)$-submodule of $Q$, see [\Lam, (8.2)].

The next proposition gives  us the common denominator property for left quotients rings.


 \re{\pripro}{Proposition {\rm [\AGR, Proposition 1.12]}} Let $R$ be a subring of an alternative ring $Q$.
\item{(i)} If $R\le_q Q$ and we take  $q_1,q_2,\dots, q_n \in Q$, with $q_1\ne 0$, then there exists $r\in N(R)$ such that $rq_1\ne 0$ and $r
q_i\in R$ for $i=1,2,\dots,n$.
\item{(ii)} Let $R\subset S\subset Q$ be three alternative rings. Then $R\le_q Q$ if and only if $R\le_q S$ and $S\le_q Q$.
\endre




 \re{\notion}{} The notion of maximal left quotient ring, in the setting of associative rings, was studied by Y. Utumi in [\U], where he proved that
rings which  are left quotient rings of themselves (equivalently rings without total right zero divisors) have a unique maximal left quotient ring.
Following the categorical definition given in [\U], we define the notion of maximal left quotient ring of an alternative ring:

 \re{\defin}{}{ \ver  Definition.} We will say that an alternative
ring $R$ has a {\it maximal left quotient ring} if there exists a
ring $Q$ such that:

\item{(i)} $Q$ is a left quotient ring of $R$ and

\item{(ii)} if $S$ is a left quotient ring of $R$ then there exists a unique monomorphism of rings $f:S\to Q$ with $f(r)=r$ for
every $r\in R$.  \endre

 Clearly, this definition implies that the maximal left quotient ring of a ring $R$, if it exists,  is unique
up to isomorphisms. We will denote it by  $Q_{max}^l(R)$.

\vfill\eject


\parrafo{Construction of the maximal left quotient ring of an alternative ring}

Let $R$ be an alternative ring. If we suppose that  $R$ has a maximal left quotient ring,
  by (\pripro(ii)) it is a left quotient ring of itself. Like in the associative case,
  we are going to prove the converse  of this fact, i.e., given an alternative ring $R$ which is a left quotient ring of
  itself we are going to prove that $R$ has a maximal left quotient ring.

 Our construction of $Q_{max}^l(R)$ (with $R$ an alternative ring) follows, surprisingly, the associative
construction, see [\Lam], where the maximal left quotient ring of an associative ring $R'$ is a quotient, under a suitable  equivalence relation, in
the set of couples $ (I, f) $, where $I$ a dense left ideal of $R'$ and $f$ is  a homomorphism  of left $R'$-modules from $I$ to $R'$, and, in
essence, the sum and the product is defined as the sum and the composition of maps.

\re{\defdenso}{}{ \ver  Definition.} We will say that a left ideal
$I$ of an alternative  ring $R$ is {\it dense} if for every
$p,q\in R$, with $p\ne 0$, there exists $a\in N(R)$ such that $
ap\ne 0$ and $aq\in I$.
\endre

This notion has similar properties to the associative one.

  \re{\Lemadenso}{Lemma {\rm {[\AGR, Lema 2.3]}}}  A left ideal $I$ of an alternative  ring $R$ is dense if and only if $R$ is a left
quotient ring of $I$.
\endre


A fundamental result in our work is that the notion of dense ideal can be characterized, essentially, by properties of the associative center of the
 ideal.



\re{\Fstar}{} We will denote by ${\cal F}^*$ the family of all left ideals $A$ of $N(R)$ with the property that for every $0\ne x\in R$ and $\mu \in
N(R)$ there exists $\lambda \in N(R)$ such that $\lambda x\ne 0$ and $\lambda \mu \in A$. We note that  $\lambda$ can be taken in $A$, and that the
intersection of a finite family of elements of ${\cal F}^*$ is an element of ${\cal F}^*$.

\re{\pFstar}{ Proposition {\rm [\AGR, Proposition 2.6]}} Let $R$ be an alternative ring and let $I$ be an ideal of $R$. Then  $I$ is a dense ideal of
$R$ if and only if $N(I)\in {\cal F}^*$.\endre



\re{\constQ}{} Let $R$ be an alternative ring. Let us construct the set that will be the maximal left quotient ring of $R$. First we define a filter
of dense left ideals: ${\cal F}:=\{R^1A \mid A\in {\cal F}^*\}$.
%\bf Remark: \rm Given $I=R^1A\in {\cal F}$ we have that $A\subset
%N(I)$, so $I=R^1A\subset R^1N(I)\subset I$. Hence, $I=R^1N(I)$,
%with $N(I)\in {\cal F}^*$.
Second we  consider the set $$S:=\{(I,f)\ \vert\ I\in {\cal F} \hbox{ and } f\in Hom^*_{N(R)}(I,R) \}$$ where $Hom^*_{N(R)}(I,R)$ denotes the set of
homomorphisms $f\in Hom_{N(R)}(I,R)$ such that for every $x\in R$ and $\lambda$, $\mu \in N(I)$ we have:

\item{(i)} $(x\lambda)f=x(\lambda)f$
\item{(ii)} $([\lambda$,$\mu])f\in N(R)$\endre

{\bf Remark:} When we define a product in $Q_{max}^l(R)$, (i) and (ii) will mean respectively that the associative center of $R$ will be contained in
the associative center of $Q_{max}^l(R)$ and the commutator of two elements of $Q_{max}^l(R)$ belongs to  its associative nucleus (which is a general
property of  alternative rings).

\re{\maxx}{} We define on $S$ the equivalence relation:
$$(I,f)\approx (I',f') \hbox{ } \Leftrightarrow \hbox{ there exists }I''\in {\cal F}\hbox{ such that } I''\subset I\cap I' \hbox{ and }
f\mid_{I''}=f'\mid_{I''}$$

We denote by $[I,f]$ the equivalence class of the element $(I,f)$ and consider the quotient $$Q:=S/\approx$$

Abusing notation, given an element $q\in Q$ we will denote by
$A_q$ any element of $\cal F^*$ and by $f_q$ any element of $
Hom^*_{N(R)}(R^1A_q,R)$ such that $q=[R^1A_q,f_q]$.

\re{\estrani}{} Let us define a  ring structure  over $Q$: If $q, q'\in Q$,  we define:

\quad $\star$ the sum: $$q+q':=[R^1(A_q\cap A_{q'})\hbox{, }f_q+f_{q'}],$$

\quad $\star$ the product:  $$qq':= [R^1A_{qq'}\hbox{, }f_{qq'}]$$

where $$A_{qq'}:=\{\lambda \in A_q \ \vert\  (\lambda)f_q\in R^1A_{q'}\} (\in {\cal F}^*), \hbox{ and}$$
$$(\sum x_ia_i)f_{qq'}:=\sum x_i((a_i)f_q)f_q'.$$

\endre

{\bf Remark:} The product is well defined and $(Q,+,\cdot)$ is a (non associative) ring. Moreover, $Q$ is unital, with $1_Q=[R^1N(R), Id_R]$ (see
[\AGR]).





\re{\alt}{Theorem \rm{[\AGR, Theorem 2.11]}} Let $Q$ be as above. Then:
\item{(i)} $R$ is a subring of $Q$. Moreover, $R$ is a dense left $N(R)$-submodule of $Q$.
 \item{(ii)} $N(R)\subset N(Q)$.
 \item{(iii)} For every $q\in Q$ and $\lambda\in N(R)$, $[\lambda,q]\in N(Q)$.
 \item{(iv)} The associator on $Q$ is an alternate  trilinear  map of its arguments over $N(Q)$.

Moreover, if $D(R)$ is 2-torsion free or semiprime,
\item{(v)}  $Q$ is an alternative ring.
\item{(vi)} $Q$ is the maximal left quotient ring of $R$.
\endre



Now, by (\pripro(ii)) and the above theorem we obtain the main theorem of the paper:

\re{\principal}{Theorem} Let $R$ be an alternative ring such that $D(R)$ is 2-torsion
free or semiprime. Then  $R$ is a left quotient ring of itself if and only if the
maximal left quotient ring of $R$ exists.
\endre


The next proposition, which is [\GSalt,  Lemma 5.7 and Proposition 6.7 (i)], shows that  the maximal ring of quotients gives us an appropriate
framework to settle the different left quotient rings that have been investigated (Fountain and Gould and classical), see [\EKI] and [\GSalt] for
definitions. This fact was used by P. Ahn and L. Marki to give a general theory of Fountain and Gould left orden in the setting of associative rings.

 \re{\mmm}{Proposition \rm{[\GSalt,(5.7) and (6.7)(i)]}} Let $R$ be an alternative ring. If $R$ is a classical
(Fountain and Gould) left order in an alternative ring $S$, then $S$ is a left quotient ring of $R$. Therefore  $S$ is a subring of $Q_{max}^l(R)$.
\endre



Some examples of maximal left quotient rings are the following ones:

 \re{\ejemplo}{}{ \ver  Examples:}

  (1) The maximal left quotient ring of an associative ring is its maximal
left quotient ring as an alternative ring.

(2) Let $Q$ be a Cayley-Dickson algebra over its center. Then $Q_{max}^l(Q)=Q$.

(3) If $R$ is a Cayley-Dickson ring, its maximal left quotient ring is a Cayley-Dickson algebra.

(4) Let us consider  a family $\{R_\alpha\}$ of alternative rings such that for every $\alpha$ there exists the maximal left quotient ring of
$R_\alpha$, which we denote by $Q_\alpha$. Then, $Q_{max}^l(\oplus R_\alpha)$ exists and it is equal to $\Pi Q_\alpha$, the direct product of the
$Q_\alpha$.

(5) Let $R$ be a nondegenerate and purely alternative ring. Then the nearly classical localization of $R$, given in [\BeMik, section 1] is the
maximal left quotient ring of $R$.
\parrafo{  References:}


[\AMI] {\ver P.N. \'Anh and L. M\'arki}, ``Left orders in regular rings with minimum condition for principal one-sided ideals". {\it  Math. Proc.
Cambridge Philos. Soc.} {\bf 109} (1991), 323--333.

[\AGR] {\ver L. Artacho, M. G\'{o}mez Lozano and G. Ruiz Calvi\~{n}o}, ``The Maximal Left Quotient Ring of an Alternative Ring". Posted on
http://mathematik. uibk.ac.at/mathematik/jordan/

[\BeMik] {\ver K.L. Beidar and A.V. Mikhalev}, ``Structure of Nondegenerate Alternative Algebras".  {\it  J. Soviet Math.} {\bf 47(3)} (1989),
2525--2536.

[\EKI] {\ver H. Essannouni and A. Kaidi} ``Goldie's Theorem for alternative rings". {\it Proc.  Amer. Math. Soc.} {\bf 121(1)} (1994), 39--45.

[\FGI] {\ver J. Fountain  and  V. Gould},  ``Orders in rings without identity" {\it Comm. Algebra} {\bf 18(9)} (1990), 3085--3110.



[\GSalt] {\ver  M. G\'omez Lozano and M. Siles Molina}, ``Left quotient rings of alternative rings". {\it Comm. Algebra} (to appear).  Posted on
http://mathematik.uibk. ac.at/mathematik/jordan/;

%[\JI] {\ver N. Jacobson}, {\it Structure and Representations  of Jordan Algebras}.  Ame\-ri\-can Mathematical Society
%Colloquium Publications, vol 39, American Mathematical Society, Providence, RI, 1968.

[\Lam] {\ver T.Y. Lam}, {\it Lectures on Modules and Rings}, Springer-Verlag, New
York, 1998.

%[\Lambek] {\ver J. Lambek}, {\it Lectures on Rings and Modules}, Chelsea Publishing Company, New York, 1976.



[\U] {\ver Y. Utumi} ``On Quotient Rings".  {\it Osaka Math. J.} {\bf  8(1)} (1956), 1--18.


[\ZSSS] {\ver    K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov and A.I. Shirshov},
 {\it Rings that are nearly associative},  Academic Press, New York, 1982.

\end



\end
