The last post I made had a little terminology mix-up, but I liked reading the responses nevertheless.
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Rings are usually assumed to have a multiplicative identity element, but sometimes this is not part of the definition of a ring. (Sometimes rings without identity elements are called “rngs.”) An example is the set of even integers (whose identity element *would* be 1, but 1 is not in the set). Or the set of sequences converging to 0, with pointwise addition and multiplication. The identity would be (1,1,1,…), but that’s not in the set. Fields always have a multiplicative identity.
Rings are not necessarily commutative, so it is not necessarily true that xy = yx for all x, y in the ring. For example, the set of n by n matrices (with elementwise addition and normal matrix multiplication) is not commutative. Fields are always commutative.
Rings may contain zero divisors, nonzero elements x so that xy = 0 for some y. Fields can’t contain zero divisors.
Rings can have proper ideals, fields can’t. That means rings can admit quotients, but fields can’t. Fields are much more rigid.
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