Before I ask my question some standard definitions (or rather my understanding of these definitions):
Let be a ring of polynomials over a field .
i.e. is a set of symbols of the form
where and is just a symbol with no meaning whatsoever.
To make it into a ring we define + and . for . These definitions are nothing but syntax rules to manipulate these symbols (using the + and . which are already defined for the Field, )
Under this definition we can write polynomial equations like
Here all we mean is that when you apply the syntax rules for + and . defined on you end up with same symbols on both sides of the equation. We assign no meaning to these symbols.
Now we define as an element of some algebric stucture where along with + and . we also define where and . . Under this, a polynomial evaluates into an element in the algebric structure . (e.g. of such a structures are: field itself or a vector space over )
My question is why should the LHS and RHS of a polynomial equation (as presented above and others as well) evaluate to a same element in ? What are the properties of and/or that make this happen?
Not sure if this is relevant question or something too trivial. But I guess it is better to ask. Any pointers would be welcome.