Theorems · Theorem · commutative algebra
Commute.isNilpotent_mul_left
∀ {R : Type u_1} {x y : R} [inst : Semiring R], Commute x y → IsNilpotent y → IsNilpotent (x * y)- Defined in
- Mathlib.RingTheory.Nilpotent.Defs
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses propext
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Commutestatement and proof · cited by 639
- IsNilpotentstatement and proof · cited by 248
- Commute.eqproof · cited by 91
- Commute.symmproof · cited by 79
- Commute.isNilpotent_mul_rightproof · cited by 5
Cited by7
Results whose statement or proof uses this declaration.
- Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotentproof · cited by 2
- Polynomial.isNilpotent_aeval_sub_of_isNilpotent_subproof · cited by 2
- PowerSeries.HasSubst.mul_rightproof · cited by 1
- LinearMap.trace_comp_eq_mul_of_commute_of_isNilpotentproof · cited by 1
- MvPowerSeries.IsNilpotent_substproof · cited by 1
- Commute.isNilpotent_mul_left_iffproof · cited by 1
- Polynomial.existsUnique_nilpotent_sub_and_aeval_eq_zeroproof · cited by 1