Theorems · Theorem · ring theory
Ideal.single_mem_jacobson_matrix
∀ {R : Type u_1} [inst : Ring R] {n : Type u_2} [inst_1 : Fintype n] [inst_2 : DecidableEq n] (I : Ideal R),
∀ x ∈ I.jacobson, ∀ (i j : n), Matrix.single i j x ∈ (Ideal.matrix n I).jacobsonA standard basis matrix is in $J(Mₙ(I))$ as long as its one possibly non-zero entry is in $J(I)$.
- Defined in
- Mathlib.LinearAlgebra.Matrix.Ideal
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingFintypeDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites46
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Finsetproof · cited by 13,712
- AddCommMonoidproof · cited by 12,281
- SetLike.coeproof · cited by 8,199
- Fintypestatement and proof · cited by 7,736
- Ringstatement and proof · cited by 7,463
- Finset.sumproof · cited by 5,195
- Idealstatement and proof · cited by 4,748
- Matrixstatement and proof · cited by 4,303
- mul_oneproof · cited by 3,885
- Finset.univproof · cited by 3,473
- one_mulproof · cited by 2,841
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.matrix_jacobson_leproof · cited by 0