The definitions of 'ring' and 'field' are pretty straightforward. For a ring (e.g. integers):
As mentioned in the comments, the term 'ring' came from Hilbert (previous question). Bill's answer there is excellent.
Fields are a bit funnier. It started with Dedekind using the word "Zahlenkörper" (body of numbers). In a supplement that he wrote to Dirichlet's Vorlesungenueber Zahlentheorie, he used that word (looking for an image for that) instead of 'rationally known quantities.' It was Moore who coined 'field' - in 1893, he wrote on Galois fields for the Bulletin of the New York Mathematical Society. However, he was always careful to mention 'fields of order' something, as field had the additional meaning at the time of neighborhood (as we understand it today).
I believe he used 'body of numbers' because they stay together just as a group does. But the transition to 'field' is a harder one to understand. Perhaps he meant it as an extension of the neighborhood-field idea - these numbers stayed in a certain neighborhood of a set of numbers in the sense that they were only a couple of powers of s away (referring to his passage below). Or perhaps he meant in to be similar to a 'body', in the sense that they just stay together.
In any case, if there were a strong, direct metaphor for 'ring' or 'field,' it is no longer readily apparent. The origins of the terms are difficult to track and harder to understand completely, and different people used the same terms in different ways. It is even possible that little thought was given to the naming of these things - it is most often hard to judge the importance of something new. Reaching for the term 'ring' or 'field' because it fits and doesn't already mean something else might be the real origin.
- addition is commutative
(1+2=2+1) - addition and multiplication are associative
(2+(2+2))=((2+2)+2) - multiplication distributes over addition
(2∗(5+7)=2∗5+2∗7) - each element has an additive inverse
(2+−2=0) - there exists an additive identity
(2+0=2)
As mentioned in the comments, the term 'ring' came from Hilbert (previous question). Bill's answer there is excellent.
Fields are a bit funnier. It started with Dedekind using the word "Zahlenkörper" (body of numbers). In a supplement that he wrote to Dirichlet's Vorlesungenueber Zahlentheorie, he used that word (looking for an image for that) instead of 'rationally known quantities.' It was Moore who coined 'field' - in 1893, he wrote on Galois fields for the Bulletin of the New York Mathematical Society. However, he was always careful to mention 'fields of order' something, as field had the additional meaning at the time of neighborhood (as we understand it today).
I believe he used 'body of numbers' because they stay together just as a group does. But the transition to 'field' is a harder one to understand. Perhaps he meant it as an extension of the neighborhood-field idea - these numbers stayed in a certain neighborhood of a set of numbers in the sense that they were only a couple of powers of s away (referring to his passage below). Or perhaps he meant in to be similar to a 'body', in the sense that they just stay together.
In any case, if there were a strong, direct metaphor for 'ring' or 'field,' it is no longer readily apparent. The origins of the terms are difficult to track and harder to understand completely, and different people used the same terms in different ways. It is even possible that little thought was given to the naming of these things - it is most often hard to judge the importance of something new. Reaching for the term 'ring' or 'field' because it fits and doesn't already mean something else might be the real origin.
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