I never learnt ring theory and am having trouble understanding the terms "valuation, kernel, residue, ideal" as applied to groups, fields and rings. Let me ask the question this way.

1) Let xt be the product of two fields. x is real x\in R and t is a member of another ordered field t\in K that has no intersection with the reals other than the number 0. Suppose that there is a mapping v that maps xt to x. Could v be called a "valuation"? If not, what is a valuation? If so, what is its kernel, residue and ideal?

2) Let f=\sum_{g\in G} x_g t_g where G is a well-ordered abelian group. x_g\in R and t_g\in K. As before, the only element that R and K have in common is the number 0. Multiplication and addition are exactly as for polynomials. This time let v be a mapping from f to the real numbers, in this case the real number that multiplies the largest t_g. Could v be called a "valuation"? If not, what is a valuation? If so, what is its kernel, residue and ideal?

Is there any good reference book on this?