TITLE: Right and left inverse of an operation in a set
QUESTION [1 upvotes]: An operation $R$ is a right inverse of the operation $S$ in a set $A$ if for any elements $x,y,z∈A$ we have $x=yRz$ if, and only if, $y=zSx$

From this, for example multiplication is the right inverse of division (since $x=z·y$ if, and only if, $y=x÷z$) but in what set? $\mathbb R-\{0\}$ right?

REPLY [0 votes]: Multiplication and division being inverse operations works on $K\setminus\{0\}$, where $K$ is any field. So, for example
$$
\mathbb Q\setminus\{0\},\ \mathbb R\setminus\{0\},\ \mathbb C\setminus\{0\},\ \mathbb F_{p^n}\setminus\{0\},\ \dots
$$
{"set_name": "stack_exchange", "score": 1, "question_id": 3552458}