Groups, rings and domains are all sets that meet certain conditions, which can be large or small, countable or uncountable. One element can be a group, "0", three elements can also be "0, 1,-1", countable: polynomials with integral coefficients (which can be verified or rings), and of course R can also be used; Ring is just another operation based on addition of commutative law and group, and the condition of field is stronger (except 0 yuan invertibility). The common ones are generally number fields, that is, integers, rational numbers, real numbers and complex numbers.

In fact, the so-called multiplication over rings and fields is not necessarily the usual multiplication. I believe there should be examples in your book. We call it multiplication.

That's all. You should want to know some concrete examples. The definition should be clear.