Theorems · Theorem · commutative algebra
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul
∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] (I : Ideal R)
(N : Submodule R M), N.FG → N ≤ I • N → ∃ r ∈ I, ∀ n ∈ N, r • n = n- Defined in
- Mathlib.RingTheory.Finiteness.Nakayama
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Submodulestatement and proof · cited by 7,192
- Idealstatement and proof · cited by 4,748
- one_smulproof · cited by 1,374
- neg_subproof · cited by 272
- Submodule.FGstatement and proof · cited by 230
- sub_smulproof · cited by 97
- Submodule.neg_memproof · cited by 31
- Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smulproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.isIdempotentElem_iff_of_fgproof · cited by 5
- Ideal.mem_iInf_smul_pow_eq_bot_iffproof · cited by 2