Hi taeto,
Thank you for kind response.
I am not a mathematician, so I am not sure how to respond to the question based on ring.
I am a physical chemist and I have some problem with me for which I am trying to develop a model using matrices. And my model requires that multiplication between two parameters be non-commutative and therefore I have chosen matrices. And to preserve the uniqueness in the result of matrix multiplication between tow matrices, I have chosen the multiplication to be outer product.
Therefore, it is I who defined x⊗y = b and y⊗x = c. And the matrices x