Trivial Square-Zero Extension

Given a ring R together with an (R, R)-bimodule M, the trivial square-zero extension of M over R is defined to be the R-algebra R ⊕ M with multiplication given by (r₁ + m₁) * (r₂ + m₂) = r₁ r₂ + r₁ m₂ + m₁ r₂.

It is a square-zero extension because M^2 = 0.

Note that expressing this requires bimodules; we write these in general for a not-necessarily-commutative R as:

variables {R M : Type*} [Semiring R] [AddCommMonoid M]
variables [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]

If we instead work with a commutative R' acting symmetrically on M, we write

variables {R' M : Type*} [CommSemiring R'] [AddCommMonoid M]
variables [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M]

noting that in this context IsCentralScalar R' M implies SMulCommClass R' R'ᵐᵒᵖ M.

Many of the later results in this file are only stated for the commutative R' for simplicity.

Main definitions

• TrivSqZeroExt.inl, TrivSqZeroExt.inr: the canonical inclusions into TrivSqZeroExt R M.
• TrivSqZeroExt.fst, TrivSqZeroExt.snd: the canonical projections from TrivSqZeroExt R M.
• triv_sq_zero_ext.algebra: the associated R-algebra structure.
• TrivSqZeroExt.lift: the universal property of the trivial square-zero extension; algebra morphisms TrivSqZeroExt R M →ₐ[R] A are uniquely defined by linear maps M →ₗ[R] A for which the product of any two elements in the range is zero.

def TrivSqZeroExt (R : Type u) (M : Type v) : Type (max u v)

"Trivial Square-Zero Extension".

Given a module M over a ring R, the trivial square-zero extension of M over R is defined to be the R-algebra R × M with multiplication given by (r₁ + m₁) * (r₂ + m₂) = r₁ r₂ + r₁ m₂ + r₂ m₁.

It is a square-zero extension because M^2 = 0.

def TrivSqZeroExt.inl {R : Type u} {M : Type v} [Zero M] (r : R) : TrivSqZeroExt R M

The canonical inclusion R → TrivSqZeroExt R M.

def TrivSqZeroExt.inr {R : Type u} {M : Type v} [Zero R] (m : M) : TrivSqZeroExt R M

The canonical inclusion M → TrivSqZeroExt R M.

def TrivSqZeroExt.fst {R : Type u} {M : Type v} (x : TrivSqZeroExt R M) : R

The canonical projection TrivSqZeroExt R M → R.

def TrivSqZeroExt.snd {R : Type u} {M : Type v} (x : TrivSqZeroExt R M) : M

The canonical projection TrivSqZeroExt R M → M.

@[simp]

theorem TrivSqZeroExt.fst_mk {R : Type u} {M : Type v} (r : R) (m : M) : TrivSqZeroExt.fst (r, m) = r

@[simp]

theorem TrivSqZeroExt.snd_mk {R : Type u} {M : Type v} (r : R) (m : M) : TrivSqZeroExt.snd (r, m) = m

theorem TrivSqZeroExt.ext {R : Type u} {M : Type v} {x : TrivSqZeroExt R M} {y : TrivSqZeroExt R M} (h1 : TrivSqZeroExt.fst x = TrivSqZeroExt.fst y) (h2 : TrivSqZeroExt.snd x = TrivSqZeroExt.snd y) : x = y

@[simp]

theorem TrivSqZeroExt.fst_inl {R : Type u} (M : Type v) [Zero M] (r : R) : TrivSqZeroExt.fst (TrivSqZeroExt.inl r) = r

@[simp]

theorem TrivSqZeroExt.snd_inl {R : Type u} (M : Type v) [Zero M] (r : R) : TrivSqZeroExt.snd (TrivSqZeroExt.inl r) = 0

@[simp]

theorem TrivSqZeroExt.fst_comp_inl {R : Type u} (M : Type v) [Zero M] : TrivSqZeroExt.fst ∘ TrivSqZeroExt.inl = id

@[simp]

theorem TrivSqZeroExt.snd_comp_inl {R : Type u} (M : Type v) [Zero M] : TrivSqZeroExt.snd ∘ TrivSqZeroExt.inl = 0

@[simp]

theorem TrivSqZeroExt.fst_inr (R : Type u) {M : Type v} [Zero R] (m : M) : TrivSqZeroExt.fst (TrivSqZeroExt.inr m) = 0

@[simp]

theorem TrivSqZeroExt.snd_inr (R : Type u) {M : Type v} [Zero R] (m : M) : TrivSqZeroExt.snd (TrivSqZeroExt.inr m) = m

@[simp]

theorem TrivSqZeroExt.fst_comp_inr (R : Type u) {M : Type v} [Zero R] : TrivSqZeroExt.fst ∘ TrivSqZeroExt.inr = 0

@[simp]

theorem TrivSqZeroExt.snd_comp_inr (R : Type u) {M : Type v} [Zero R] : TrivSqZeroExt.snd ∘ TrivSqZeroExt.inr = id

theorem TrivSqZeroExt.inl_injective {R : Type u} {M : Type v} [Zero M] : Function.Injective TrivSqZeroExt.inl

theorem TrivSqZeroExt.inr_injective {R : Type u} {M : Type v} [Zero R] : Function.Injective TrivSqZeroExt.inr

Structures inherited from Prod

Additive operators and scalar multiplication operate elementwise.

instance TrivSqZeroExt.inhabited {R : Type u} {M : Type v} [Inhabited R] [Inhabited M] : Inhabited (TrivSqZeroExt R M)

instance TrivSqZeroExt.zero {R : Type u} {M : Type v} [Zero R] [Zero M] : Zero (TrivSqZeroExt R M)

instance TrivSqZeroExt.add {R : Type u} {M : Type v} [Add R] [Add M] : Add (TrivSqZeroExt R M)

instance TrivSqZeroExt.sub {R : Type u} {M : Type v} [Sub R] [Sub M] : Sub (TrivSqZeroExt R M)

instance TrivSqZeroExt.neg {R : Type u} {M : Type v} [Neg R] [Neg M] : Neg (TrivSqZeroExt R M)

instance TrivSqZeroExt.addSemigroup {R : Type u} {M : Type v} [AddSemigroup R] [AddSemigroup M] : AddSemigroup (TrivSqZeroExt R M)

instance TrivSqZeroExt.addZeroClass {R : Type u} {M : Type v} [AddZeroClass R] [AddZeroClass M] : AddZeroClass (TrivSqZeroExt R M)

instance TrivSqZeroExt.addMonoid {R : Type u} {M : Type v} [AddMonoid R] [AddMonoid M] : AddMonoid (TrivSqZeroExt R M)

instance TrivSqZeroExt.addGroup {R : Type u} {M : Type v} [AddGroup R] [AddGroup M] : AddGroup (TrivSqZeroExt R M)

instance TrivSqZeroExt.addCommSemigroup {R : Type u} {M : Type v} [AddCommSemigroup R] [AddCommSemigroup M] : AddCommSemigroup (TrivSqZeroExt R M)

instance TrivSqZeroExt.addCommMonoid {R : Type u} {M : Type v} [AddCommMonoid R] [AddCommMonoid M] : AddCommMonoid (TrivSqZeroExt R M)

instance TrivSqZeroExt.addCommGroup {R : Type u} {M : Type v} [AddCommGroup R] [AddCommGroup M] : AddCommGroup (TrivSqZeroExt R M)

instance TrivSqZeroExt.smul {S : Type u_2} {R : Type u} {M : Type v} [SMul S R] [SMul S M] : SMul S (TrivSqZeroExt R M)

instance TrivSqZeroExt.isScalarTower {T : Type u_1} {S : Type u_2} {R : Type u} {M : Type v} [SMul T R] [SMul T M] [SMul S R] [SMul S M] [SMul T S] [IsScalarTower T S R] [IsScalarTower T S M] : IsScalarTower T S (TrivSqZeroExt R M)

instance TrivSqZeroExt.smulCommClass {T : Type u_1} {S : Type u_2} {R : Type u} {M : Type v} [SMul T R] [SMul T M] [SMul S R] [SMul S M] [SMulCommClass T S R] [SMulCommClass T S M] : SMulCommClass T S (TrivSqZeroExt R M)

instance TrivSqZeroExt.isCentralScalar {S : Type u_2} {R : Type u} {M : Type v} [SMul S R] [SMul S M] [SMul Sᵐᵒᵖ R] [SMul Sᵐᵒᵖ M] [IsCentralScalar S R] [IsCentralScalar S M] : IsCentralScalar S (TrivSqZeroExt R M)

instance TrivSqZeroExt.mulAction {S : Type u_2} {R : Type u} {M : Type v} [Monoid S] [MulAction S R] [MulAction S M] : MulAction S (TrivSqZeroExt R M)

instance TrivSqZeroExt.distribMulAction {S : Type u_2} {R : Type u} {M : Type v} [Monoid S] [AddMonoid R] [AddMonoid M] [DistribMulAction S R] [DistribMulAction S M] : DistribMulAction S (TrivSqZeroExt R M)

instance TrivSqZeroExt.module {S : Type u_2} {R : Type u} {M : Type v} [Semiring S] [AddCommMonoid R] [AddCommMonoid M] [Module S R] [Module S M] : Module S (TrivSqZeroExt R M)

@[simp]

theorem TrivSqZeroExt.fst_zero {R : Type u} {M : Type v} [Zero R] [Zero M] : TrivSqZeroExt.fst 0 = 0

@[simp]

theorem TrivSqZeroExt.snd_zero {R : Type u} {M : Type v} [Zero R] [Zero M] : TrivSqZeroExt.snd 0 = 0

@[simp]

theorem TrivSqZeroExt.fst_add {R : Type u} {M : Type v} [Add R] [Add M] (x₁ : TrivSqZeroExt R M) (x₂ : TrivSqZeroExt R M) : TrivSqZeroExt.fst (x₁ + x₂) = TrivSqZeroExt.fst x₁ + TrivSqZeroExt.fst x₂

@[simp]

theorem TrivSqZeroExt.snd_add {R : Type u} {M : Type v} [Add R] [Add M] (x₁ : TrivSqZeroExt R M) (x₂ : TrivSqZeroExt R M) : TrivSqZeroExt.snd (x₁ + x₂) = TrivSqZeroExt.snd x₁ + TrivSqZeroExt.snd x₂

@[simp]

theorem TrivSqZeroExt.fst_neg {R : Type u} {M : Type v} [Neg R] [Neg M] (x : TrivSqZeroExt R M) : TrivSqZeroExt.fst (-x) = -TrivSqZeroExt.fst x

@[simp]

theorem TrivSqZeroExt.snd_neg {R : Type u} {M : Type v} [Neg R] [Neg M] (x : TrivSqZeroExt R M) : TrivSqZeroExt.snd (-x) = -TrivSqZeroExt.snd x

@[simp]

theorem TrivSqZeroExt.fst_sub {R : Type u} {M : Type v} [Sub R] [Sub M] (x₁ : TrivSqZeroExt R M) (x₂ : TrivSqZeroExt R M) : TrivSqZeroExt.fst (x₁ - x₂) = TrivSqZeroExt.fst x₁ - TrivSqZeroExt.fst x₂

@[simp]

theorem TrivSqZeroExt.snd_sub {R : Type u} {M : Type v} [Sub R] [Sub M] (x₁ : TrivSqZeroExt R M) (x₂ : TrivSqZeroExt R M) : TrivSqZeroExt.snd (x₁ - x₂) = TrivSqZeroExt.snd x₁ - TrivSqZeroExt.snd x₂

@[simp]

theorem TrivSqZeroExt.fst_smul {S : Type u_2} {R : Type u} {M : Type v} [SMul S R] [SMul S M] (s : S) (x : TrivSqZeroExt R M) : TrivSqZeroExt.fst (s • x) = s • TrivSqZeroExt.fst x

@[simp]

theorem TrivSqZeroExt.snd_smul {S : Type u_2} {R : Type u} {M : Type v} [SMul S R] [SMul S M] (s : S) (x : TrivSqZeroExt R M) : TrivSqZeroExt.snd (s • x) = s • TrivSqZeroExt.snd x

theorem TrivSqZeroExt.fst_sum {R : Type u} {M : Type v} {ι : Type u_3} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → TrivSqZeroExt R M) : TrivSqZeroExt.fst (Finset.sum s fun i => f i) = Finset.sum s fun i => TrivSqZeroExt.fst (f i)

theorem TrivSqZeroExt.snd_sum {R : Type u} {M : Type v} {ι : Type u_3} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → TrivSqZeroExt R M) : TrivSqZeroExt.snd (Finset.sum s fun i => f i) = Finset.sum s fun i => TrivSqZeroExt.snd (f i)

@[simp]

theorem TrivSqZeroExt.inl_zero {R : Type u} (M : Type v) [Zero R] [Zero M] : TrivSqZeroExt.inl 0 = 0

@[simp]

theorem TrivSqZeroExt.inl_add {R : Type u} (M : Type v) [Add R] [AddZeroClass M] (r₁ : R) (r₂ : R) : TrivSqZeroExt.inl (r₁ + r₂) = TrivSqZeroExt.inl r₁ + TrivSqZeroExt.inl r₂

@[simp]

theorem TrivSqZeroExt.inl_neg {R : Type u} (M : Type v) [Neg R] [SubNegZeroMonoid M] (r : R) : TrivSqZeroExt.inl (-r) = -TrivSqZeroExt.inl r

@[simp]

theorem TrivSqZeroExt.inl_sub {R : Type u} (M : Type v) [Sub R] [SubNegZeroMonoid M] (r₁ : R) (r₂ : R) : TrivSqZeroExt.inl (r₁ - r₂) = TrivSqZeroExt.inl r₁ - TrivSqZeroExt.inl r₂

@[simp]

theorem TrivSqZeroExt.inl_smul {S : Type u_2} {R : Type u} (M : Type v) [Monoid S] [AddMonoid M] [SMul S R] [DistribMulAction S M] (s : S) (r : R) : TrivSqZeroExt.inl (s • r) = s • TrivSqZeroExt.inl r

theorem TrivSqZeroExt.inl_sum {R : Type u} (M : Type v) {ι : Type u_3} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → R) : TrivSqZeroExt.inl (Finset.sum s fun i => f i) = Finset.sum s fun i => TrivSqZeroExt.inl (f i)

@[simp]

theorem TrivSqZeroExt.inr_zero (R : Type u) {M : Type v} [Zero R] [Zero M] : TrivSqZeroExt.inr 0 = 0

@[simp]

theorem TrivSqZeroExt.inr_add (R : Type u) {M : Type v} [AddZeroClass R] [AddZeroClass M] (m₁ : M) (m₂ : M) : TrivSqZeroExt.inr (m₁ + m₂) = TrivSqZeroExt.inr m₁ + TrivSqZeroExt.inr m₂

@[simp]

theorem TrivSqZeroExt.inr_neg (R : Type u) {M : Type v} [SubNegZeroMonoid R] [Neg M] (m : M) : TrivSqZeroExt.inr (-m) = -TrivSqZeroExt.inr m

@[simp]

theorem TrivSqZeroExt.inr_sub (R : Type u) {M : Type v} [SubNegZeroMonoid R] [Sub M] (m₁ : M) (m₂ : M) : TrivSqZeroExt.inr (m₁ - m₂) = TrivSqZeroExt.inr m₁ - TrivSqZeroExt.inr m₂

@[simp]

theorem TrivSqZeroExt.inr_smul {S : Type u_2} (R : Type u) {M : Type v} [Zero R] [Zero S] [SMulWithZero S R] [SMul S M] (r : S) (m : M) : TrivSqZeroExt.inr (r • m) = r • TrivSqZeroExt.inr m

theorem TrivSqZeroExt.inr_sum (R : Type u) {M : Type v} {ι : Type u_3} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → M) : TrivSqZeroExt.inr (Finset.sum s fun i => f i) = Finset.sum s fun i => TrivSqZeroExt.inr (f i)

theorem TrivSqZeroExt.inl_fst_add_inr_snd_eq {R : Type u} {M : Type v} [AddZeroClass R] [AddZeroClass M] (x : TrivSqZeroExt R M) : TrivSqZeroExt.inl (TrivSqZeroExt.fst x) + TrivSqZeroExt.inr (TrivSqZeroExt.snd x) = x

theorem TrivSqZeroExt.ind {R : Type u_3} {M : Type u_4} [AddZeroClass R] [AddZeroClass M] {P : TrivSqZeroExt R M → Prop} (h : (r : R) → (m : M) → P (TrivSqZeroExt.inl r + TrivSqZeroExt.inr m)) (x : TrivSqZeroExt R M) : P x

To show a property hold on all TrivSqZeroExt R M it suffices to show it holds on terms of the form inl r + inr m.

This can be used as induction x using TrivSqZeroExt.ind.

theorem TrivSqZeroExt.linearMap_ext {S : Type u_2} {R : Type u} {M : Type v} {N : Type u_3} [Semiring S] [AddCommMonoid R] [AddCommMonoid M] [AddCommMonoid N] [Module S R] [Module S M] [Module S N] ⦃f : TrivSqZeroExt R M →ₗ[S] N⦄ ⦃g : TrivSqZeroExt R M →ₗ[S] N⦄ (hl : ∀ (r : R), ↑f (TrivSqZeroExt.inl r) = ↑g (TrivSqZeroExt.inl r)) (hr : ∀ (m : M), ↑f (TrivSqZeroExt.inr m) = ↑g (TrivSqZeroExt.inr m)) : f = g

This cannot be marked @[ext] as it ends up being used instead of LinearMap.prod_ext when working with R × M.

@[simp]

theorem TrivSqZeroExt.inrHom_apply (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (m : M) : ↑(TrivSqZeroExt.inrHom R M) m = TrivSqZeroExt.inr m

def TrivSqZeroExt.inrHom (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] : M →ₗ[R] TrivSqZeroExt R M

The canonical R-linear inclusion M → TrivSqZeroExt R M.

@[simp]

theorem TrivSqZeroExt.sndHom_apply (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (x : TrivSqZeroExt R M) : ↑(TrivSqZeroExt.sndHom R M) x = TrivSqZeroExt.snd x

def TrivSqZeroExt.sndHom (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] : TrivSqZeroExt R M →ₗ[R] M

The canonical R-linear projection TrivSqZeroExt R M → M.

Multiplicative structure

instance TrivSqZeroExt.one {R : Type u} {M : Type v} [One R] [Zero M] : One (TrivSqZeroExt R M)

instance TrivSqZeroExt.mul {R : Type u} {M : Type v} [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] : Mul (TrivSqZeroExt R M)

@[simp]

theorem TrivSqZeroExt.fst_one {R : Type u} {M : Type v} [One R] [Zero M] : TrivSqZeroExt.fst 1 = 1

@[simp]

theorem TrivSqZeroExt.snd_one {R : Type u} {M : Type v} [One R] [Zero M] : TrivSqZeroExt.snd 1 = 0

@[simp]

theorem TrivSqZeroExt.fst_mul {R : Type u} {M : Type v} [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] (x₁ : TrivSqZeroExt R M) (x₂ : TrivSqZeroExt R M) : TrivSqZeroExt.fst (x₁ * x₂) = TrivSqZeroExt.fst x₁ * TrivSqZeroExt.fst x₂

@[simp]

theorem TrivSqZeroExt.snd_mul {R : Type u} {M : Type v} [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] (x₁ : TrivSqZeroExt R M) (x₂ : TrivSqZeroExt R M) : TrivSqZeroExt.snd (x₁ * x₂) = TrivSqZeroExt.fst x₁ • TrivSqZeroExt.snd x₂ + MulOpposite.op (TrivSqZeroExt.fst x₂) • TrivSqZeroExt.snd x₁

@[simp]

theorem TrivSqZeroExt.inl_one {R : Type u} (M : Type v) [One R] [Zero M] : TrivSqZeroExt.inl 1 = 1

@[simp]

theorem TrivSqZeroExt.inl_mul {R : Type u} (M : Type v) [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r₁ : R) (r₂ : R) : TrivSqZeroExt.inl (r₁ * r₂) = TrivSqZeroExt.inl r₁ * TrivSqZeroExt.inl r₂

theorem TrivSqZeroExt.inl_mul_inl {R : Type u} (M : Type v) [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r₁ : R) (r₂ : R) : TrivSqZeroExt.inl r₁ * TrivSqZeroExt.inl r₂ = TrivSqZeroExt.inl (r₁ * r₂)

@[simp]

theorem TrivSqZeroExt.inr_mul_inr (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] (m₁ : M) (m₂ : M) : TrivSqZeroExt.inr m₁ * TrivSqZeroExt.inr m₂ = 0

theorem TrivSqZeroExt.inl_mul_inr {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] (r : R) (m : M) : TrivSqZeroExt.inl r * TrivSqZeroExt.inr m = TrivSqZeroExt.inr (r • m)

theorem TrivSqZeroExt.inr_mul_inl {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] (r : R) (m : M) : TrivSqZeroExt.inr m * TrivSqZeroExt.inl r = TrivSqZeroExt.inr (MulOpposite.op r • m)

instance TrivSqZeroExt.mulOneClass {R : Type u} {M : Type v} [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] : MulOneClass (TrivSqZeroExt R M)

instance TrivSqZeroExt.addMonoidWithOne {R : Type u} {M : Type v} [AddMonoidWithOne R] [AddMonoid M] : AddMonoidWithOne (TrivSqZeroExt R M)

@[simp]

theorem TrivSqZeroExt.fst_nat_cast {R : Type u} {M : Type v} [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : TrivSqZeroExt.fst ↑n = ↑n

@[simp]

theorem TrivSqZeroExt.snd_nat_cast {R : Type u} {M : Type v} [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : TrivSqZeroExt.snd ↑n = 0

@[simp]

theorem TrivSqZeroExt.inl_nat_cast {R : Type u} {M : Type v} [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : TrivSqZeroExt.inl ↑n = ↑n

instance TrivSqZeroExt.addGroupWithOne {R : Type u} {M : Type v} [AddGroupWithOne R] [AddGroup M] : AddGroupWithOne (TrivSqZeroExt R M)

@[simp]

theorem TrivSqZeroExt.fst_int_cast {R : Type u} {M : Type v} [AddGroupWithOne R] [AddGroup M] (z : ℤ) : TrivSqZeroExt.fst ↑z = ↑z

@[simp]

theorem TrivSqZeroExt.snd_int_cast {R : Type u} {M : Type v} [AddGroupWithOne R] [AddGroup M] (z : ℤ) : TrivSqZeroExt.snd ↑z = 0

@[simp]

theorem TrivSqZeroExt.inl_int_cast {R : Type u} {M : Type v} [AddGroupWithOne R] [AddGroup M] (z : ℤ) : TrivSqZeroExt.inl ↑z = ↑z

instance TrivSqZeroExt.nonAssocSemiring {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] : NonAssocSemiring (TrivSqZeroExt R M)

instance TrivSqZeroExt.nonAssocRing {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] : NonAssocRing (TrivSqZeroExt R M)

instance TrivSqZeroExt.instPowTrivSqZeroExtNat {R : Type u} {M : Type v} [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] : Pow (TrivSqZeroExt R M) ℕ

In the general non-commutative case, the power operator is

$$\begin{align} (r + m)^n &= r^n + r^{n-1}m + r^{n-2}mr + \cdots + rmr^{n-2} + mr^{n-1} \ & =r^n + \sum_{i = 0}^{n - 1} r^{(n - 1) - i} m r^{i} \end{align}$$

In the commutative case this becomes the simpler $(r + m)^n = r^n + nr^{n-1}m$.

@[simp]

theorem TrivSqZeroExt.fst_pow {R : Type u} {M : Type v} [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (x : TrivSqZeroExt R M) (n : ℕ) : TrivSqZeroExt.fst (x ^ n) = TrivSqZeroExt.fst x ^ n

theorem TrivSqZeroExt.snd_pow_eq_sum {R : Type u} {M : Type v} [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (x : TrivSqZeroExt R M) (n : ℕ) : TrivSqZeroExt.snd (x ^ n) = List.sum (List.map (fun i => TrivSqZeroExt.fst x ^ (Nat.pred n - i) • MulOpposite.op (TrivSqZeroExt.fst x ^ i) • TrivSqZeroExt.snd x) (List.range n))

theorem TrivSqZeroExt.snd_pow_of_smul_comm {R : Type u} {M : Type v} [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] (x : TrivSqZeroExt R M) (n : ℕ) (h : MulOpposite.op (TrivSqZeroExt.fst x) • TrivSqZeroExt.snd x = TrivSqZeroExt.fst x • TrivSqZeroExt.snd x) : TrivSqZeroExt.snd (x ^ n) = n • TrivSqZeroExt.fst x ^ Nat.pred n • TrivSqZeroExt.snd x

@[simp]

theorem TrivSqZeroExt.snd_pow {R : Type u} {M : Type v} [CommMonoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [IsCentralScalar R M] (x : TrivSqZeroExt R M) (n : ℕ) : TrivSqZeroExt.snd (x ^ n) = n • TrivSqZeroExt.fst x ^ Nat.pred n • TrivSqZeroExt.snd x

@[simp]

theorem TrivSqZeroExt.inl_pow {R : Type u} {M : Type v} [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (n : ℕ) : TrivSqZeroExt.inl r ^ n = TrivSqZeroExt.inl (r ^ n)

instance TrivSqZeroExt.monoid {R : Type u} {M : Type v} [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Monoid (TrivSqZeroExt R M)

theorem TrivSqZeroExt.fst_list_prod {R : Type u} {M : Type v} [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] (l : List (TrivSqZeroExt R M)) : TrivSqZeroExt.fst (List.prod l) = List.prod (List.map TrivSqZeroExt.fst l)

instance TrivSqZeroExt.semiring {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Semiring (TrivSqZeroExt R M)

theorem TrivSqZeroExt.snd_list_prod {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] (l : List (TrivSqZeroExt R M)) : TrivSqZeroExt.snd (List.prod l) = List.sum (List.map (fun x => List.prod (List.take x.fst (List.map TrivSqZeroExt.fst l)) • MulOpposite.op (List.prod (List.drop (Nat.succ x.fst) (List.map TrivSqZeroExt.fst l))) • TrivSqZeroExt.snd x.snd) (List.enum l))

The second element of a product $\prod_{i=0}^n (r_i + m_i)$ is a sum of terms of the form $r_0\cdots r_{i-1}m_ir_{i+1}\cdots r_n$.

instance TrivSqZeroExt.ring {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Ring (TrivSqZeroExt R M)

instance TrivSqZeroExt.commMonoid {R : Type u} {M : Type v} [CommMonoid R] [AddCommMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [IsCentralScalar R M] : CommMonoid (TrivSqZeroExt R M)

instance TrivSqZeroExt.commSemiring {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] : CommSemiring (TrivSqZeroExt R M)

instance TrivSqZeroExt.commRing {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] : CommRing (TrivSqZeroExt R M)

@[simp]

theorem TrivSqZeroExt.inlHom_apply (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] (r : R) : ↑(TrivSqZeroExt.inlHom R M) r = TrivSqZeroExt.inl r

def TrivSqZeroExt.inlHom (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] : R →+* TrivSqZeroExt R M

The canonical inclusion of rings R → TrivSqZeroExt R M.

instance TrivSqZeroExt.algebra' (S : Type u_1) (R : Type u) (M : Type v) [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] : Algebra S (TrivSqZeroExt R M)

instance TrivSqZeroExt.instAlgebraTrivSqZeroExtSemiringToSemiringOp_rightToSMulToZeroToAddMonoidToSMulZeroClassToZeroToMonoidWithZeroToSMulWithZeroToMulActionWithZeroToZeroToCommMonoidWithZeroMulOppositeZeroMonoidWithZeroSemiringSmulCommClass_selfToCommMonoidToMulAction (R' : Type u) (M : Type v) [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] : Algebra R' (TrivSqZeroExt R' M)

theorem TrivSqZeroExt.algebraMap_eq_inl (R' : Type u) (M : Type v) [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] : ↑(algebraMap R' (TrivSqZeroExt R' M)) = TrivSqZeroExt.inl

theorem TrivSqZeroExt.algebraMap_eq_inlHom (R' : Type u) (M : Type v) [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] : algebraMap R' (TrivSqZeroExt R' M) = TrivSqZeroExt.inlHom R' M

theorem TrivSqZeroExt.algebraMap_eq_inl' (S : Type u_1) (R : Type u) (M : Type v) [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] (s : S) : ↑(algebraMap S (TrivSqZeroExt R M)) s = TrivSqZeroExt.inl (↑(algebraMap S R) s)

@[simp]

theorem TrivSqZeroExt.fstHom_apply (S : Type u_1) (R : Type u) (M : Type v) [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] (x : TrivSqZeroExt R M) : ↑(TrivSqZeroExt.fstHom S R M) x = TrivSqZeroExt.fst x

def TrivSqZeroExt.fstHom (S : Type u_1) (R : Type u) (M : Type v) [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] : TrivSqZeroExt R M →ₐ[S] R

The canonical R-algebra projection TrivSqZeroExt R M → R.

theorem TrivSqZeroExt.algHom_ext {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] ⦃f : TrivSqZeroExt R' M →ₐ[R'] A⦄ ⦃g : TrivSqZeroExt R' M →ₐ[R'] A⦄ (h : ∀ (m : M), ↑f (TrivSqZeroExt.inr m) = ↑g (TrivSqZeroExt.inr m)) : f = g

theorem TrivSqZeroExt.algHom_ext' {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] ⦃f : TrivSqZeroExt R' M →ₐ[R'] A⦄ ⦃g : TrivSqZeroExt R' M →ₐ[R'] A⦄ (h : LinearMap.comp (AlgHom.toLinearMap f) (TrivSqZeroExt.inrHom R' M) = LinearMap.comp (AlgHom.toLinearMap g) (TrivSqZeroExt.inrHom R' M)) : f = g

def TrivSqZeroExt.liftAux {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] (f : M →ₗ[R'] A) (hf : ∀ (x y : M), ↑f x * ↑f y = 0) : TrivSqZeroExt R' M →ₐ[R'] A

There is an alg_hom from the trivial square zero extension to any R-algebra with a submodule whose products are all zero.

See TrivSqZeroExt.lift for this as an equiv.

@[simp]

theorem TrivSqZeroExt.liftAux_apply_inr {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] (f : M →ₗ[R'] A) (hf : ∀ (x y : M), ↑f x * ↑f y = 0) (m : M) : ↑(TrivSqZeroExt.liftAux f hf) (TrivSqZeroExt.inr m) = ↑f m

@[simp]

theorem TrivSqZeroExt.liftAux_comp_inrHom {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] (f : M →ₗ[R'] A) (hf : ∀ (x y : M), ↑f x * ↑f y = 0) : LinearMap.comp (AlgHom.toLinearMap (TrivSqZeroExt.liftAux f hf)) (TrivSqZeroExt.inrHom R' M) = f

@[simp]

theorem TrivSqZeroExt.liftAux_inrHom {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] : TrivSqZeroExt.liftAux (TrivSqZeroExt.inrHom R' M) (_ : ∀ (m₁ m₂ : M), TrivSqZeroExt.inr m₁ * TrivSqZeroExt.inr m₂ = 0) = AlgHom.id R' (TrivSqZeroExt R' M)

@[simp]

theorem TrivSqZeroExt.lift_symm_apply_coe {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] (F : TrivSqZeroExt R' M →ₐ[R'] A) : ↑(↑TrivSqZeroExt.lift.symm F) = LinearMap.comp (AlgHom.toLinearMap F) (TrivSqZeroExt.inrHom R' M)

@[simp]

theorem TrivSqZeroExt.lift_apply {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] (f : { f // ∀ (x y : M), ↑f x * ↑f y = 0 }) : ↑TrivSqZeroExt.lift f = TrivSqZeroExt.liftAux ↑f (_ : ∀ (x y : M), ↑↑f x * ↑↑f y = 0)

def TrivSqZeroExt.lift {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] : { f // ∀ (x y : M), ↑f x * ↑f y = 0 } ≃ (TrivSqZeroExt R' M →ₐ[R'] A)

A universal property of the trivial square-zero extension, providing a unique TrivSqZeroExt R M →ₐ[R] A for every linear map M →ₗ[R] A whose range has no non-zero products.

This isomorphism is named to match the very similar Complex.lift.