Theorems · Theorem · commutative algebra
Module.nontrivial
∀ (R : Type u_5) (M : Type u_6) [inst : MonoidWithZero R] [Nontrivial M] [inst_2 : Zero M] [MulActionWithZero R M], Nontrivial R
A semiring is Nontrivial provided that there exists a nontrivial module over this semiring.
- Defined in
- Mathlib.Algebra.Module.Defs
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Nontrivialstatement and proof · cited by 2,416
- MonoidWithZerostatement and proof · cited by 456
- MulActionWithZerostatement and proof · cited by 79
- MulActionWithZero.nontrivialproof · cited by 3
Cited by17
Results whose statement or proof uses this declaration.
- AlgebraicIndependent.extendScalars_of_isIntegralproof · cited by 2
- LinearMap.det_restrictScalarsproof · cited by 2
- Algebra.PowerBasis.norm_gen_eq_prod_rootsproof · cited by 2
- Polynomial.div_prod_eq_quo_add_sum_rem_divproof · cited by 2
- Module.rank_pos_of_freeproof · cited by 2
- IsTranscendenceBasis.of_isAlgebraic_adjoin_insert_sdiffproof · cited by 2
- IsIntegral.trans_isAlgebraicproof · cited by 1
- cardinalMk_algHom_le_rankproof · cited by 1
- LinearMap.equivOfIsUnitDet_applyproof · cited by 1
- Polynomial.quo_add_sum_rem_div_uniqueproof · cited by 1
- Subalgebra.rank_botproof · cited by 1