Add a few more ZZIdl methods

src/AlgAss/AbsAlgAss.jl

@@ -1212,7 +1212,7 @@ end
 @doc raw"""
      radical(A::AbstractAssociativeAlgebra) -> AbsAlgAssIdl
 
-Returns the Jacobson-Radical of $A$.
+Returns the Jacobson radical of $A$.
 """
 function radical(A::AbstractAssociativeAlgebra{T}) where { T } #<: Union{ fpFieldElem, EuclideanRingResidueFieldElem{ZZRingElem}, fqPolyRepFieldElem, FqPolyRepFieldElem, QQFieldElem, AbsSimpleNumFieldElem } }
   return ideal_from_gens(A, _radical(A), :twosided)

src/NumField/QQ.jl

@@ -127,7 +127,13 @@ end
 
 *(x::ZZIdl, y::ZZIdl) = ZZIdl(x.gen * y.gen)
 
-intersect(x::ZZIdl, y::ZZIdl) = ZZIdl(lcm(x.gen, y.gen))
+function intersect(x::ZZIdl, y::ZZIdl...)
+  g = gen(x)
+  for I in y
+    g = lcm(g, gen(I))
+  end
+  return ZZIdl(g)
+end
 
 lcm(x::ZZIdl, y::ZZIdl) = intersect(x, y)
 
@@ -152,6 +158,13 @@ isone(I::ZZIdl) = isone(I.gen)
 iszero(I::ZZIdl) = iszero(gen(I))
 is_maximal(I::ZZIdl) = is_prime(gen(I))
 is_prime(I::ZZIdl) = is_zero(I) || is_maximal(I)
+is_primary(I::ZZIdl) = is_zero(I) || is_prime_power_with_data(gen(I))[1]
+
+is_subset(I::ZZIdl, J::ZZIdl) = is_divisible_by(gen(J), gen(I))
+
+radical(I::ZZIdl) = iszero(I) ? I : ideal(ZZ, radical(gen(I)))
+primary_decomposition(I::ZZIdl) = iszero(I) ? [ (I,I) ] :
+  [ (ideal(ZZ, p^k), ideal(ZZ, p)) for (p,k) in factor(gen(I)) ]
 
 maximal_order(::QQField) = ZZ
 

test/NumField/QQ.jl

@@ -1,3 +1,19 @@
+function check_primary_decomposition(J::Any)
+  @testset "Testing primary decomposition for $J" begin
+    d = primary_decomposition(J)
+    @test all(is_primary(Q) for (Q,P) in d)
+    @test all(is_prime(P) for (Q,P) in d)
+    @test all(is_subset(P,Q) for (Q,P) in d)
+    if isempty(d)
+      @test isone(J)
+      @test J == radical(J)
+    else
+      @test intersect([Q for (Q,P) in d]...) == J
+      @test prod([P for (Q,P) in d]) == radical(J)
+    end
+  end
+end
+
 @testset "NumField/QQ" begin
   @test Hecke.ideal_type(ZZ) == Hecke.ZZIdl
   @test Hecke.ideal_type(Hecke.ZZRing) == Hecke.ZZIdl
@@ -19,6 +35,8 @@
   @test !is_one(I)
   @test is_maximal(I)
   @test is_prime(I)
+  @test is_primary(I)
+  check_primary_decomposition(I)
 
   J = 4*ZZ
 
@@ -31,18 +49,35 @@
   @test !is_one(J)
   @test !is_maximal(J)
   @test !is_prime(J)
+  @test is_primary(J)
+  check_primary_decomposition(J)
 
   J = 0*ZZ
   @test is_zero(J)
   @test !is_one(J)
   @test !is_maximal(J)
   @test is_prime(J)
+  @test is_primary(J)
+  check_primary_decomposition(J)
 
   J = 1*ZZ
   @test !is_zero(J)
   @test is_one(J)
   @test !is_maximal(J)
   @test !is_prime(J)
+  @test !is_primary(J)
+  check_primary_decomposition(J)
+
+  J = 36*ZZ
+  @test !is_zero(J)
+  @test !is_one(J)
+  @test !is_maximal(J)
+  @test !is_prime(J)
+  @test !is_primary(J)
+  @test radical(J) == 6*ZZ
+  check_primary_decomposition(J)
+
+  check_primary_decomposition((2*3*5)^2*ZZ)
 
   I = QQ(1, 2)*ZZ
   @test I ==  ZZ * QQ(1, 2)