Let C be any category. A set of 2elements of an object X is called invertible if it generates 1_{X} . A proper ideal of X is called maximal if it is not contained in any other proper ideal of X. Recall that an object with exactly two ideals is called a simple object. Remark 3.1. (a) Suppose X is an
object such that the product X
X exits. Then the 2element (p_{1}, p_{2}):
X X > X of projections
is invertible.
Proposition 3.2. (a) Any maximal ideal
is a 2kernel.
Proof: (a) Any ideal is an intersection of 2kernels.
Thus any maximal ideal must be a 2kernel.
Definition 3.3. (a) An ideal a
of an object X is called radical
if for any 2element r Ï a
there exists an arrow t: X > Z such that t(r)
is invertible and t(a) is noninvertible.
Proposition 3.4. (a) Any maximal ideal
is radical.
Proof. (a) This follows from (3.2).
Proposition 3.5. (a) Intersections of
radical ideals are radical.
Proof. (a) First note that 1_{X}
is always radical. Suppose a is the intersection of a set {a_{i}}
of radical ideals of X. We may assume a is proper. If a 2arrow
r is not in a, then r is not in a_{i}
for some i. Since a_{i} is radical we can
find an arrow t: X > Z such that t(r) is
invertible and t(a_{i}) is not. Since a
a_{i}, t(a) is noninvertible.
This shows that a is radical.
For any set S of 2elements of an object X we denote by (S) the set of 2 elements r of X such that, if t: X > Z is an arrow and t(r) is invertible, then t(S) is invertible. Let r(S) be the intersection of all the radical ideals containing S. It follows from (3.1.b) that (S) = ((S)) and r(S) = r((S)). Remark 3.6. If {a_{i}} is a set of ideals X then r( a_{i}) = r( r(a_{i})). Denote by I^{r}(X) the set of radical ideals of X. This means that r: I(X) ® I^{r}(X) preserves joins. Remark 3.7. Suppose a and b
are two ideals of X.
Definition 3.8. A unitary category is called radical if any nonterminal object has a proper radical ideal. Suppose C is a radical category. Proposition 3.9. Suppose a is a
proper ideal of an object X and t: X > Z is an arrow.
Then
Proof. (a) Since a is proper it is contained in a proper
kernel ker(t) for an arrow t: X > Z such
that Z is nonnull. Since C is radical Z has a proper
radical ideal b. Then t^{1}(b) is a proper
radical ideal containing a.
Corollary 3.10. If a is an ideal of an object X then r(a) = 1 if and only if a = 1. Proof. This follows from (3.9.a) that any proper ideal is contained in a proper radical ideal. Proposition 3.11. Suppose a and b are two proper
ideals of an object X. Then the following conditions are equivalent:
Proof. (a) and (b) are equivalent by (3.9.c) and (3.7.e). The assertions (b) and (c) are equivalent by (3.9.d). Proposition 3.12. A unitary category C is reduced if and only if any ideal of an object is radical. Proof. If the condition holds then the ideal 0_{X}
for any object X is radical, which implies that C is reduced.
Conversely assume C is a reduced category. Then any 2 kernel
of an object is radical by (3.5.b). Since any ideal
is an intersection of 2kernels, it is radical by (3.5.a).
