Theorems · Theorem · commutative algebra
Algebra.trace_algebraMap
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [StrongRankCondition R]
[Module.Free R S] (x : R), (Algebra.trace R S) ((algebraMap R S) x) = Module.finrank R S • xIf x is in the base field K, then the trace is [L : K] * x.
(If L is not finite-dimensional over K, then trace and finrank return 0.)
- Defined in
- Mathlib.RingTheory.Trace.Defs
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- RingHom.idstatement · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- Finsetproof · cited by 13,712
- Algebrastatement and proof · cited by 11,388
- LinearMapstatement · cited by 10,215
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement and proof · cited by 4,706
- Module.finrankstatement and proof · cited by 1,770
- Module.Basisproof · cited by 1,477
- Fintype.cardproof · cited by 1,386
- Module.Freestatement and proof · cited by 597
Cited by3
Results whose statement or proof uses this declaration.
- trace_eq_trace_adjoinproof · cited by 3
- IsCyclotomicExtension.discr_prime_powproof · cited by 3
- Algebra.trace_eq_zero_of_not_isSeparableproof · cited by 2