I'll explain a bit of the background of this topic first.

One of the common algebras we use is a field . Because it's a field, the + and • operations within the real numbers have to follow a set amount of axioms. Some of these axioms define the existence and interactions of inverse elements, which effectively fill the roles of subtraction and division, so we don't have to make an algebraic structure (it won't be a field anymore) like this .

Your last question is essentially asking whether is a field. In this algebraic structure subtraction and division are the main operations and addition and multiplication are derived from them. If this structure is a field, it's going to work just as well as does. So, we have to check the field axioms and whether they work. Sadly, your structure fails both commutativity axioms (probably even others as well): for any elements a, b which are of R, it must hold that a-b=b-a and a÷b=b÷a. This is obviously not true for most values of a and b, therefore it's not a field. You could still use the structure and have it follow some list of axioms to make it work the same, however, it would be quite a non-standart way of approaching the problem and not worth the hassle.