ROCKY MOUNTAIN
JOURNAL OF MATHEMATICS
Volume 2, Number 4, Fall 1972
QUOTIENT CATEGORIES AND RINGS OF QUOTIENTS 12
CAROL L. WALKER AND ELBERT A. WALKER
A Serre class in an abelian category cA is a nonempty subclass <£ of
<A closed under subobjects, quotient objects, and extensions. The
importance of such classes stems from the fact that it is for such classes
S that the quotient category cAIS is defined [4]. Quotient categories
provide the natural categorical setting for certain considerations.
(See, for example, [4] and [16].)
In studying the categories cRlS, with <R the category of (left) R
modules and <£ a Serre class of J?, the opposite ring R^ of the endomorphism ring of R as an object of <R\£ plays a fundamental role.
The ring R^ has many properties reminiscent of "rings of quotients,"
and R± is examined from this point of view in §2. In particular, each
of the various generalizations of rings of quotients known to the
authors is an R^ for a suitable <£, and several examples are given. The
modules M^ are also discussed, and it is indicated how these objects
provide a reasonable generalization of rings and modules of quotients.
This point of view unifies previous generalizations and places them
in a natural categorical setting. §2 concludes with an examination of
Rs in the case where R is a commutative Noetherian ring and S is the
Serre class of modules with essential socles.
In [3], Gabriel has investigated the quotient categories cRlS, where
Jt is the category of all (left) modules over a ring R, and S is a Serre
class closed under arbitrary infinite direct sums (a strongly complete
Serre class). There is a canonical functor, called the localization
functor, from <R to <R±, the category of all left modules over Rs. Every
Rs module is an R module via a natural ring homomorphism <f> : R
—» R^, cRs fi S (the class of Rs modules which belong to <£ when
considered as R modules via $) is a Serre class of cR^, and in [ 3 ] , a
Received by the editors November 5, 1970.
AMS 1970 subject classifications. Primary 16A08; Secondary 18E35.
ir
The work on this paper was partially supported by NSF Grant GP-28379.
2
A version of this paper was submitted to another journal in January, 1964,
but for various reasons was never published. Although certain of these results
have subsequently been published by others, the Rocky Mountain Journal of
Mathematics felt that, because of its overall merit and frequent references to it
in the literature, this revised version of the paper should appear in its
entirety —Editor.
Copyright © 1972 Rocky Mountain Mathematics Consortium
513