I have tried get a version of the proof stating that a left ideals of a ring is not, in general, a right ideal, and viceversa. Is my formulation right? Comments and corrections are welcome. I have tried to understand some German papers on the topic I came across.......I hope it is fine......The proof takes the form of a reductio ad absurdum...... Thanks in advance:
The proof goes as follows: let us assume M to be a left module over the ring R with the external operation R × M → M. Then we write the product with elements from R, proceeding from right to left. For that purpose, the symbol ◦ is used. Let then α·x := x◦α, ∀α ∈ R, x ∈ M. Now, generally speaking, the left module M with the external product ◦ : M × R → M is no right module whatsoever. This contradicts the Associative Law. According to the initial assumption M is a left module, that is, it is precisely the Associative Law that comes into play for scalar multiplication of a ring element from the left side. We then suppose that M with the operation ◦ is also a right module. Assuming, without loss of generality, that all other conditions apply as usual, it can be seen that instead of the Associative Law (M3) the following holds: (αβ) · x = x ◦ (αβ) = (x ◦ α) ◦ β = (α · x) ◦ β = β · (α · x)⊥, which constitutes a contradiction to the Associativity of the left module.