algebraMap_eq_zero_iff — Mathlib

English

If R has no zero divisors, then algebraMap x = 0 in FreeAlgebra R X iff x = 0.

Русский

Если в R нет делителей нуля, то algebraMap x = 0 в FreeAlgebra R X эквивалентно x = 0.

LaTeX

$$$\text{NoZeroDivisors } R \Rightarrow (\mathrm{algebraMap}_R^{\mathrm{FreeAlgebra}}(x)=0 \Leftrightarrow x=0)$$$

Lean4

@[simp]
theorem algebraMap_eq_zero_iff (x : R) : algebraMap R (FreeAlgebra R X) x = 0 ↔ x = 0 :=
  map_eq_zero_iff (algebraMap _ _) algebraMap_leftInverse.injective

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