Categorical
Concepts in Commutative Algebra
(based on [Unitary Categories])
Consider a commutative ring
a
= ^{-1}a1 (or it generates the largest ideal of
R).
(categorical definition) An element a
of R is a unit iff for any ring homomorphism f: R
--> S, f(a) = 0 implies that S
is the zero ring 0.
n.
(categorical definition) An element a
of a ring is nilpotent iff for any ring homomorphism f: R
-->
S,
f(a) is a unit implies that S is the
zero ring 0.
R is an ideal if it is closed under the operations - and
with any element of R.
(categorical definition) A subset
of aR is an ideal if it is the kernel of a homomorphism (i.e. the
inverse image of the zero element under a ring homomorphism).
radical iff for any element a of R, a
is in ^{n} implie that aa is in .
a(categorical definition) An ideal
is aradical iff it contains any element a such that, if f:
R
--> S is any ring homomorphism with f(a) a unit, then
f()
generates the largest ideal of aS.
prime iff for any element a, b of R, ab is
in R iff a or b is in .
a(categorical definition) A non-zero ideal
is aprime iff it is radical and universal irreducible (i.e. an ideal
is airreducible if the intersection of and b
is in c implies that a or b is
in c; a. is called auniversaly irreducible
if
its inverse image under any homomorphism is irreducible).
Z[T]
--> R factors through the zero homomorphism 0:
_{Z}Z[T]
--> Z; the codiagonal homomorphism from the sum
Z[T,
T']
of Z[T] to Z[T] is the pushout of
0:
_{Z}Z[T]
--> Z along substraction operator - :
Z[T]
--> Z[T,
T'] sending
T to
T - T'.
(c) Z is a terminal object of CRing.
The homomorphism Z[T]
--> Z is a generic coequalizer in the following sense:
E
--> X. For any a:
E --> X and f:
X
--> Y write f(a) = 0 if f
= _{a}0. Next define a subset of _{Y}E(X) to be
an ideal on X if it is an intersection of the kernels f(^{-1}0)
for some maps f with domain X. Just as in the case of commutative
rings one can define the notions of radical ideals and prime ideals,
etc.
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