When we write where F is, say, a field, do we necessarily mean the set of all possible polynomials in x_1, x_2, ... ... x_n with coefficients in F? [In this case, essentially all that is required to determine whether a polynomial belongs to is to check that the co-efficients belong to F and the indeterminates only contain .]
when e write do we mean to include possible cases such as the set of polynomials with even coefficients - that is we may be talking about the set of polynomials with even co-efficients - so we cannot be sure what ring of polynomials we are talking about when we write until we specify the exact nature of ring of polynomials we are talking about further.
If the latter is the case when given we can not reason about whether particular polynomials belong to until you know the exact nature of the ring
I very much suspect that the former is the case but ... ... Can someone please confirm or clarify this.