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Theorems · Theorem · commutative algebra

IsSMulRegular.zero_iff_subsingleton

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

The element 0 is M-regular if and only if M is trivial.

Defined in
Mathlib.Algebra.Regular.SMul
Cited by
2 results in Mathlib
Foundations
Depth 9 from the axioms · uses no axioms
Assumes
MonoidWithZeroZeroMulActionWithZero
Mathlib docsSourceLoogleLeanSearch

Around this declaration

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

IsSMulRegular.not_zero_iff · cited by 1IsSMulRegular.not_zero_iffIsSMulRegular.zero · cited by 1IsSMulRegular.zeroMonoidWithZero · cited by 456MonoidWithZeroIsSMulRegular · cited by 128IsSMulRegularMulActionWithZero · cited by 79MulActionWithZeroIsSMulRegular.subsingleton · cited by 1IsSMulRegular.subsingletonIsSMulRegular.zero_iff_subsin…CITED BYCITES

Cites4

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Cited by2

Results whose statement or proof uses this declaration.