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Theorems · Theorem · commutative algebra

IsSMulRegular.not_zero_iff

∀ {R : Type u_1} {M : Type u_3} [inst : MonoidWithZero R] [inst_1 : Zero M] [inst_2 : MulActionWithZero R M],
  ¬IsSMulRegular M 0 ↔ Nontrivial M

The 0 element is not M-regular, on a non-trivial module.

Defined in
Mathlib.Algebra.Regular.SMul
Cited by
1 results in Mathlib
Foundations
Depth 14 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
MonoidWithZeroZeroMulActionWithZero
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

IsSMulRegular.not_zero · cited by 2IsSMulRegular.not_zeroNontrivial · cited by 2416NontrivialMonoidWithZero · cited by 456MonoidWithZeroIsSMulRegular · cited by 128IsSMulRegularMulActionWithZero · cited by 79MulActionWithZeronot_iff_comm · cited by 32not_iff_commnontrivial_iff · cited by 12nontrivial_iffsubsingleton_iff · cited by 10subsingleton_iffIsSMulRegular.zero_iff_subsingleton · cited by 2IsSMulRegular.zero_iff_su…IsSMulRegular.not_zero_iffCITED BYCITES

Cites8

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Cited by1

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