Examples of zero-divisors in AddMonoidAlgebras #
This file contains an easy source of zero-divisors in an AddMonoidAlgebra.
If k is a field and G is an additive group containing a non-zero torsion element, then
k[G] contains non-zero zero-divisors: this is lemma zero_divisors_of_torsion.
There is also a version for periodic elements of an additive monoid: zero_divisors_of_periodic.
The converse of this statement is Kaplansky's zero divisor conjecture.
The formalized example generalizes in trivial ways the assumptions: the field k can be any
nontrivial ring R and the additive group G with a torsion element can be any additive monoid
A with a non-zero periodic element.
Besides this example, we also address a comment in Data.Finsupp.Lex to the effect that the proof
that addition is monotone on α →₀ N uses that it is strictly monotone on N.
The specific statement is about Finsupp.Lex.covariantClass_lt_left and its analogue
Finsupp.Lex.covariantClass_le_right. We do not need two separate counterexamples, since the
operation is commutative.
The example is very simple. Let F = {0, 1} with order determined by 0 < 1 and absorbing
addition (which is the same as max in this case). We denote a function f : F → F (which is
automatically finitely supported!) by [f 0, f 1], listing its values. Recall that the order on
finitely supported function is lexicographic, matching the list notation. The inequality
[0, 1] ≤ [1, 0] holds. However, adding [1, 0] to both sides yields the reversed inequality
[1, 1] > [1, 0].
This is a simple example showing that if R is a non-trivial ring and A is an additive
monoid with an element a satisfying n • a = a and (n - 1) • a ≠ a, for some 2 ≤ n,
then R[A] contains non-zero zero-divisors. The elements are easy to write down:
[a] and [a] ^ (n - 1) - 1 are non-zero elements of R[A] whose product
is zero.
Observe that such an element a cannot be invertible. In particular, this lemma never applies
if A is a group.
This is a simple example showing that if R is a non-trivial ring and A is an additive
monoid with a non-zero element a of finite order oa, then R[A] contains
non-zero zero-divisors. The elements are easy to write down:
∑ i in Finset.range oa, [a] ^ i and [a] - 1 are non-zero elements of R[A]
whose product is zero.
In particular, this applies whenever the additive monoid A is an additive group with a non-zero
torsion element.
F is the type with two elements zero and one. We define the "obvious" linear order and
absorbing addition on it to generate our counterexample.
- zero: Counterexample.F
- one: Counterexample.F
Instances For
Equations
- Counterexample.instDecidableEqF x y = if h : Counterexample.F.toCtorIdx x = Counterexample.F.toCtorIdx y then isTrue ⋯ else isFalse ⋯
Equations
- Counterexample.instInhabitedF = { default := Counterexample.F.zero }
The same as List.getRest, except that we take the "rest" from the first match, rather than
from the beginning, returning [] if there is no match. For instance,
#eval dropUntil [1,2] [3,1,2,4,1,2] -- [4, 1, 2]
Equations
- Counterexample.List.dropUntil x [] = []
- Counterexample.List.dropUntil x (a :: as) = Option.getD (List.getRest (a :: as) x) (Counterexample.List.dropUntil x as)
Instances For
guard_decl na mod makes sure that the declaration with name na is in the module mod.
guard_decl Nat.nontrivial Mathlib.Algebra.Ring.Nat -- does nothing
guard_decl Nat.nontrivial Not.In.Here
-- the module Not.In.Here is not imported!
This test makes sure that the comment referring to this example is in the file claimed in the doc-module to this counterexample.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
- Counterexample.F.instZeroF = { zero := Counterexample.F.zero }
1 is not really needed, but it is nice to use the notation.
Equations
- Counterexample.F.instOneF = { one := Counterexample.F.one }
A tactic to prove trivial goals by enumeration.
Equations
- Counterexample.F.tacticBoom = Lean.ParserDescr.node `Counterexample.F.tacticBoom 1024 (Lean.ParserDescr.nonReservedSymbol "boom" false)
Instances For
val maps 0 1 : F to their counterparts in ℕ.
We use it to lift the linear order on ℕ.
Equations
- Counterexample.F.val x = match x with | Counterexample.F.zero => 0 | Counterexample.F.one => 1
Instances For
Equations
- Counterexample.F.instAddF = { add := max }
F would be a CommSemiring, using min as multiplication. Again, we do not need this.
The CovariantClasses asserting monotonicity of addition hold for F.
Equations
A few simp-lemmas to take care of trivialities in the proof of the example below.