split_by_characteristic_domain
theorem split_by_characteristic_domain :
∀ (R : Type u_1) [inst : CommRing R] {P : Prop} [inst_1 : IsDomain R],
(∀ (p : ℕ), p.Prime → CharP R p → P) → (Algebra ℚ R → P) → (∀ (p : ℕ), p.Prime → MixedCharZero R p → P) → P
This theorem states that given a commutative ring `R` that is also a domain, for any proposition `P`, we can split `P` over `R` into three cases:
1. The case where `R` has a prime characteristic `p` (i.e., `p` is a prime number and every integer multiple of `p` is zero in `R`). For this case, if the proposition holds for all such prime characteristics, denoted as `p.Prime → CharP R p → P`.
2. The case where `R` has characteristic zero (i.e., no positive integer is zero in `R`, similar to the characteristic of the set of rational numbers ℚ). For this case, the proposition holds if there is a ring homomorphism from the rationals to `R`, denoted as `Algebra ℚ R → P`.
3. The case where `R` has a mixed characteristic `(0, p)`. This situation occurs when `R` itself has characteristic zero, but its prime subring has prime characteristic `p`. For this case, if the proposition holds for all such mixed characteristics, denoted as `p.Prime → MixedCharZero R p → P`.
Then, under these conditions, the proposition `P` holds in general in `R`, denoted as `P`.
More concisely: Given a commutative domain `R`, a proposition `P` holds in `R` if it holds for all prime characteristics `p`, for characteristic zero with a ring homomorphism from the rationals, and for mixed characteristics `(0, p)`.
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EqualCharZero.of_not_mixedCharZero
theorem EqualCharZero.of_not_mixedCharZero :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : CharZero R],
(∀ p > 0, ¬MixedCharZero R p) → ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)
This theorem states that if a commutative ring `R` does not have mixed characteristic, then it must have equal characteristic zero. In more detail, given any type `R` that has a commutative ring structure and a characteristic zero structure, if for all positive integers `p`, `R` does not have mixed characteristic `p`, then for any ideal `I` in `R` that is not the whole ring, the quotient ring `R ⧸ I` will also have characteristic zero.
More concisely: If `R` is a commutative ring without mixed characteristic, then it has characteristic zero and any quotient ring `R ⧸ I` of an ideal `I` different from `R` also has characteristic zero.
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MixedCharZero.reduce_to_maximal_ideal
theorem MixedCharZero.reduce_to_maximal_ideal :
∀ (R : Type u_1) [inst : CommRing R] {p : ℕ},
p.Prime → ((∃ I, I ≠ ⊤ ∧ CharP (R ⧸ I) p) ↔ ∃ I, I.IsMaximal ∧ CharP (R ⧸ I) p)
This theorem, known as "Reduction to `I` prime ideal", is about mixed characteristic rings in the field of abstract algebra. It states that for any commutative ring `R` and any prime number `p`, if you have a situation where there exists an ideal `I` (a subset of `R` with certain properties) such that `I` is not the whole ring and the ring `R` modulo `I` has characteristic `p`, then you can actually assume that `I` is a maximal ideal (an ideal that has no other ideals containing it except the whole ring). This theorem is particularly useful when proving statements about mixed characteristic rings, as it allows for a reduction to a simpler scenario with a maximal ideal.
More concisely: If a commutative ring `R` and prime number `p` have an ideal `I` such that `R mod I` has characteristic `p` and `I` is not the whole ring, then `I` is maximal.
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EqualCharZero.of_algebraRat
theorem EqualCharZero.of_algebraRat :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : Algebra ℚ R] (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)
This theorem states that for any commutative ring `R` that is a rational number (`ℚ`) algebra, given an ideal `I` of `R` which is not the whole ring (`R`), the quotient ring `R/I` has characteristic zero. In other words, if you have a ring structure on `R` that extends the rational numbers, and an ideal `I` that isn't the whole ring, then there can be no non-zero natural number `n` such that `n*1 = 0` in the quotient ring `R/I`. This property is known as having characteristic zero.
More concisely: For any commutative ring `R` that is a rational number algebra and contains an ideal `I` distinct from `R`, the quotient ring `R/I` has characteristic zero.
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isEmpty_algebraRat_iff_mixedCharZero
theorem isEmpty_algebraRat_iff_mixedCharZero :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : CharZero R], IsEmpty (Algebra ℚ R) ↔ ∃ p > 0, MixedCharZero R p := by
sorry
This theorem states that for any commutative ring `R` with characteristic zero, `R` is not a rational number (`ℚ`) algebra if and only if there exists a prime number `p` greater than 0 such that `R` has a mixed characteristic with respect to `p`. In other words, a commutative ring of characteristic zero lacks a structure of `ℚ`-algebra iff it exhibits mixed characteristic for some prime number.
More concisely: A commutative ring of characteristic zero has no rational algebra structure if and only if it exhibits mixed characteristic with respect to some prime number.
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EqualCharZero.PNat.isUnit_natCast
theorem EqualCharZero.PNat.isUnit_natCast :
∀ {R : Type u_1} [inst : CommRing R] [h : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))] (n : ℕ+), IsUnit ↑↑n
This theorem, `EqualCharZero.PNat.isUnit_natCast`, asserts that for any positive natural number `n`, in the context of a `Commutative Ring` R with a characteristic zero property for all non-trivial quotient rings, the natural number `n` is a unit when considered as an element of `R`. In other words, every positive natural number has a multiplicative inverse in such a ring `R`.
More concisely: In a commutative ring R with characteristic zero, every positive natural number is a unit when considered as an element.
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EqualCharZero.nonempty_algebraRat_iff
theorem EqualCharZero.nonempty_algebraRat_iff :
∀ (R : Type u_1) [inst : CommRing R], Nonempty (Algebra ℚ R) ↔ ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)
This theorem states that for any commutative ring `R`, `R` can be considered as an algebra over the rationals (`ℚ`) if and only if for any left ideal `I` of `R` that is not the whole ring, the characteristic of the quotient ring `R/I` is zero. In other words, a ring is a rational algebra if the characteristic — the smallest positive integer `n` such that `n * 1 = 0` in the ring, or zero if no such `n` exists — of every quotient ring by a non-trivial ideal is zero.
More concisely: A commutative ring is a rational algebra if and only if the characteristic of every quotient ring by a non-zero ideal is zero.
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split_by_characteristic_localRing
theorem split_by_characteristic_localRing :
∀ (R : Type u_1) [inst : CommRing R] {P : Prop} [inst_1 : LocalRing R],
(∀ (p : ℕ), IsPrimePow p → CharP R p → P) →
(Algebra ℚ R → P) → (∀ (p : ℕ), p.Prime → MixedCharZero R p → P) → P
For any type `R` with a commutative ring structure and local ring structure, the theorem `split_by_characteristic_localRing` states that any property `P` over `R` can be divided into three cases:
- The case where the characteristic of the ring is a prime power. In this case, `P` holds if for any natural number `p`, if `p` is a prime power and `R` has characteristic `p`, then `P` holds.
- The case where the characteristic of the ring is zero. In this case, `P` holds if there exists a ring homomorphism from the field of rational numbers to `R`.
- The case of mixed characteristic `(0, p)`. In this case, `P` holds if for any natural number `p`, if `p` is prime and `R` has mixed characteristic zero and `p`, then `P` holds.
In other words, the theorem provides a way to consider every characteristic of a local ring when proving a property about it.
More concisely: For any property `P` over a commutative ring `R` with a local ring structure, `P` holds if it holds for prime power characteristics, for characteristic zero with a ring homomorphism from the rational numbers, or for mixed characteristic zero and a prime characteristic.
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EqualCharZero.to_not_mixedCharZero
theorem EqualCharZero.to_not_mixedCharZero :
∀ (R : Type u_1) [inst : CommRing R], (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)) → ∀ p > 0, ¬MixedCharZero R p
This theorem states that if a commutative ring `R` has the property that for all of its left ideals `I` (other than the whole ring itself), the characteristic of the quotient ring `R ⧸ I` is zero, then `R` does not have mixed characteristic for any positive integer `p`. In other words, it cannot be the case that the characteristic of `R` is a positive integer `p` and the characteristic of its residual field is zero.
More concisely: If a commutative ring `R` has the property that the quotient ring `R ⧸ I` has zero characteristic for all proper left ideals `I`, then `R` has characteristic zero.
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split_by_characteristic
theorem split_by_characteristic :
∀ (R : Type u_1) [inst : CommRing R] {P : Prop},
(∀ (p : ℕ), p ≠ 0 → CharP R p → P) → (Algebra ℚ R → P) → (∀ (p : ℕ), p.Prime → MixedCharZero R p → P) → P
The theorem `split_by_characteristic` provides a method for splitting any property `P` over a commutative ring `R` into three distinct cases based on the ring's characteristic. The three cases are:
1. The case of positive characteristic, which is defined for all prime numbers `p` different from zero such that the ring `R` has characteristic `p`.
2. The case of equal characteristic zero, which is defined when `R` is an algebra over the field of rational numbers `ℚ`.
3. The case of mixed characteristic `(0, p)`, which is defined for all prime numbers `p` such that the ring `R` has mixed characteristic zero `p`.
The theorem states that if property `P` holds true for all three cases, then property `P` holds true in general.
More concisely: If a property `P` holds for all commutative rings `R` in their respective cases of positive characteristic `p` different from zero, equal characteristic zero as an algebra over `ℚ`, and mixed characteristic `(0, p)`, then property `P` holds for `R` in general.
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EqualCharZero.pnatCast_eq_natCast
theorem EqualCharZero.pnatCast_eq_natCast :
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))] (n : ℕ+),
↑↑n = ↑↑n
This theorem states that for any positive integer `n`, given any commutative ring `R` with the property that for any ideal `I` in the ring that is not the whole ring, the characteristic of the quotient ring `R/I` is zero (`CharZero`), the double cast of `n` to a positive integer and then to a natural number is equal to itself. Essentially, it's saying that the casting process doesn't change the number `n`.
More concisely: For any commutative ring `R` with the property that the characteristic of every proper ideal is zero, the double cast of any positive integer `n` to a positive integer and then to a natural number equals `n`.
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EqualCharZero.pnatCast_one
theorem EqualCharZero.pnatCast_one :
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))], ↑1 = 1
This theorem states that for any commutative ring `R` with the property that for every ideal `I` in `R` which is not equal to the whole ring, `R` modulo `I` is a characteristic zero ring, the natural number cast of `1` is equal to `1`. Essentially, this is asserting that in such a commutative ring, casting the natural number `1` to a ring element results in the ring's unit element.
More concisely: For any commutative ring `R` such that the quotient `R / I` is of characteristic zero for every non-unit ideal `I`, the natural number `1` maps to the ring's multiplicative identity in `R`.
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EqualCharZero.iff_not_mixedCharZero
theorem EqualCharZero.iff_not_mixedCharZero :
∀ (R : Type u_1) [inst : CommRing R] [inst_1 : CharZero R],
(∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)) ↔ ∀ p > 0, ¬MixedCharZero R p
This theorem states that for any commutative ring `R` with characteristic zero, `R` is said to have equal characteristic zero if and only if it does not have mixed characteristic for any positive integer `p`. Specifically, if for all ideals `I` in `R` that are not equal to the entire ring, the quotient ring `R ⧸ I` has characteristic zero, this is equivalent to the assertion that for all positive integers `p`, the ring `R` does not have a mixed characteristic `p`. In this context, a ring is said to have characteristic zero if the multiplicative identity (1) of the ring, added to itself any number of times, never results in the additive identity (0). A ring has mixed characteristic `(0,p)` if its underlying additive group has characteristic zero but its additive ideal generated by `1` has characteristic `p`.
More concisely: A commutative ring with characteristic zero has no non-trivial ideals generating mixed characteristics, if and only if every such ideal quotient has characteristic zero.
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MixedCharZero.reduce_to_p_prime
theorem MixedCharZero.reduce_to_p_prime :
∀ (R : Type u_1) [inst : CommRing R] {P : Prop},
(∀ p > 0, MixedCharZero R p → P) ↔ ∀ (p : ℕ), p.Prime → MixedCharZero R p → P
The theorem `MixedCharZero.reduce_to_p_prime` states that in the context of proving any proposition `P` about mixed characteristic rings, it is sufficient to consider only the cases where `p` is a prime number. More specifically, if one can establish the truth of `P` for all positive integer `p` that give the ring `R` a mixed characteristic of `p`, then `P` is also true for all prime `p` that give `R` a mixed characteristic. Conversely, if `P` is true for all prime `p` that give `R` a mixed characteristic, then `P` is true for all positive integer `p` that give `R` a mixed characteristic.
More concisely: In the context of mixed characteristic rings, proving a proposition `P` for all prime numbers `p` that give the ring a mixed characteristic is equivalent to proving it for all positive integers `p` with the same property.
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split_equalCharZero_mixedCharZero
theorem split_equalCharZero_mixedCharZero :
∀ (R : Type u_1) [inst : CommRing R] {P : Prop} [inst_1 : CharZero R],
(Algebra ℚ R → P) → (∀ (p : ℕ), p.Prime → MixedCharZero R p → P) → P
This theorem states that for any commutative ring `R` with characteristic zero, any property `P` holds if and only if it holds in two specific scenarios. The first scenario is when `R` has a rational number algebra structure. The second scenario is for every prime natural number `p`, the ring `R` has a mixed characteristic zero `p` structure. The theorem essentially splits a property in a ring of characteristic zero into the case of equal characteristic (with rational numbers) and the case of mixed characteristic.
More concisely: For any commutative ring R with characteristic zero, a property P holds if and only if it holds in R with a rational number algebra structure and for every prime natural number p, R has a mixed characteristic zero p structure.
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