Ring came from Zahlring, which was Hilbert's term for what we would essentially call a ring of algebraic integers. Dedekind earlier used the term ordnung (= order, taken from the Linnean classification terminology like class and genus). For more on this, see the comments to the question http://mathoverflow.net/questions/17062/why-is-h-the-notation-for-class-numbers.
Fields in the algebraic sense used to be called bodies (thus closer to French and German). [Edit: In 1900, Pierpoint's "Galois' Theory of Algebraic Equations, Part II", in the second volume of Annals of Mathematics, uses "body" for field and "inferior body" for subfield, introduced on page 25. In 1910, Legh Reid's "The elements of the theory of algebraic numbers" uses the term "realm" for field, or more specifically for number field. Reid's text can be found on Google books, and on p. vi of the preface he writes that "realm" is synonymous with Körper, corpus, campus, body, domain, and field. In 1934, Heilbronn and Linfoot wrote a paper "On the imaginary quadratic corpora of class-number one", so corpus was still in use in the early 1930s.]