Classical Objects
Zhaohua Luo
(11/18/1998)
(a draft)
Y. Diers in [Diers,
1992, p.45] defined "a range of objects corresponding to the usual types
of commutative ring" for a Zariski category (cf. [Archive]).
In this note we show how to define these classical objects for any right
unitary category.
Let C be a category.
Definition 1. (a) A difference
of an object X is a notation a  b where (a,
b)
is any pair of parallel morphisms from an object to X; a  b
is a zero difference
if a = b.
(b) A morphism t: X > T
is called a solution
of X (or any difference a  b of X); if
T
is
a terminal object then we say that t is a trivial
solution of X.
Definition 2. (a) A solution t of
a difference a  b is a zero solution
if ta = tb.
(b) A difference a  b of an object is unit
if
its zero solutions are trivial.
(c) A solution t of a difference a
 b is a unit solution
if ta  tb is a unit.
(d) A difference a  b is nilpotent
if its unit solutions are trivial.
(e) A difference a  b is regular
if it has a monomorphic solution.
Remark 3. Suppose a  b is a difference and t
is a solution of a  b of an object X.
(a) If a  b is both a unit and a zero then X
is terminal.
(b) a  b is a zero (resp. unit, resp. nilpotent) implies that
ta
 tb is so.
In the following we assume any object of C has a unit.
Definition 4. (a) An object is reduced
if any nonzero difference has a nontrivial unit solution (i.e. it has
no nonzero nilpotent difference).
(b) A nonterminal object is integral
if
any two nonzero differences has a common monomorphic unit solution (i.e.
any nonzero difference is regular).
(c) A nonterminal object is primary
if
any two nonnilpotent differences has a common monomorphic unit solution.
(d) A nonterminal object is quasiprimary
if
any two nonnilpotent differences has a common nontrivial
unit solution.
(e) A nonterminal object is simple
(i.e.
a field)
if any nonzero difference is a unit.
(f) A nonterminal object is pseudosimple
if any nonnilpotent difference is a unit.
(g) A nonterminal object is
local
if
the class of nonunit differences has a common noninitial null solution.
(h) A nonterminal object is
generic
if
the class of nonnilpotent differences has a common noninitial unit solution.
Remark 5. (a) The classes of reduced, integral, primary,
quasiprimary objects are closed under subobjects.
(b) An object is integral iff it is reduced and primary.
(c) Any simple object is integral; any subobject of a simple object
is integral.
(d) Any pseudosimple is primary and any primary object is quasiprimary;
any subobject of a pseudosimple object is primary..
(e) An object is simple iff it is reduced and a pseudosimple.
Remark 6. Suppose U = hom(W, ~) is a representable
faithful functor from C to the category of sets. By a Udifference
(or Wdifference) of an object X we mean a notation a
 b, where (a,
b) is any pair of elements of U(X).
Applying the above methods we obtain the notion of Ureduced, Uprime
objects, etc. with respective to Udifferences. One can show that
an object is reduced (resp. prime, etc.) if it is Ureduced (resp.
Uprime,
etc.) Thus for a concrete category (C, U) (e.g. an algebraic
geometry) it suffices to consider Udifferences instead of general
differences.
Theorem 7. Suppose (C, U) is an algebraic
geometry.
(a) An object is integral iff any nonzero difference has a monomorphic
unit solution (or any two nonzero differences has a common nontrivial
unit solution).
(b) An object is primary iff any nonnilpotent difference has a monomorphic
unit solution (or any two nonnilpotent differences has a common nontrivial
unit solution)..
(c) An object is integral iff it is reduced and quasiprimary.
(d) An object is integral iff it is a subobject of a simple object.
(e) An object is primary iff it is a subobject of a pseudosimple object.
(f) Any direct product of reduced object is reduced; an object is reduced
iff it is a subobject of a direct product of simple objects (see [Categorical
Geometry, Chapter 3  5] for the proof) .
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