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I considered solving of the system of linear equations a¹₁ₒ₀x¹a¹₁ₒ₁+. . . +a¹₍ₒ₀xⁿa¹₍ₒ₁=b¹. . . aⁿ₁ₒ₀x¹aⁿ₁ₒ₁+. . . +aⁿ₍ₒ₀xⁿaⁿ₍ₒ₁=bⁿ over non-commutative associative algebra. I considered examples in quaternion algebra. I considered also Newton's method to solve the equation f (x) =a over non-commutative associative algebra.
Aleks Kleyn (2024) studied this question.