A ring is a set R equipped with 2 binary operations, which might be denoted by any symbols, but which I will denote by + and *, and containing 2 special elements 0 and 1, such that the following rules hold for all elements a,b,c of R: (associativity of +) (a+b)+c = a+(b+c) (commutativity of +) a+b = b+a (identity for +) a+0 = a (inverses for +) there exists an element -a in R with a+(-a) = 0 (associativity of *) (a*b)*c = a*(b*c) (identity for *) a*1 = a = 1*a (distributivity of * over +) Left: a*(b+c) = (a*b)+(a*c) Right: (a+b)*c = (a*c)+(b*c) Those are the basic axioms. A ring is called commutative if * is also commutative, i.e., if a*b = b*a for all elements a,b of R. The set of integers {...,-2,-1,0,1,2,...} with the usual operations of addition (+) and multiplication (*) forms a commutative ring, and it is the most basic example. An interesting example studied by C.F. Gauss around 1800 (because of his interests in number theory) is the commutative ring of Gaussian integers, namely, all complex numbers a+bi such that a,b are integers; again the operations are addition and multiplication. An example of a non-commutative ring is given by the set of all 2x2 matrices with integer entries, with matrix addition and matrix multiplication as operations. (Of course one can use n x n matrices for any positive integer n.) That's the definition, but so what? Well, examples of rings are quite common, but many of them, such as the Gaussian integers, are concocted to aid in the study of something else. Thus as in any area of mathematics, the study of rings has external motivations (the study of rings introduced to solve some other problem) and internal motivations (once one begins to study rings, natural questions and natural classes start to arise). In both cases, more and more conditions are imposed to develop deeper and deeper theories. [A main point of abstract algebra is to treat as many different problems as possible simultaneously by recognizing them as the manifestation of a single abstract system; however, the more sophisticated the problems we tackle become, the more additional conditions and definitions we need in order to solve them.] The most important condition is probably the Noetherian condition, a technical condition introduced in the 20th century (used in disguise before it was formally defined), named after Emmy Noether, a pioneering worker on commutative rings. The biggest division in the subject is between commutative rings and non-commutative ones. Commutative rings arose in number theory (as in the Gaussian integers) and were also used in the study of polynomials. These days commutative rings are tied very closely to the theory of algebraic geometry, the study of curves, surfaces, etc., defined by polynomial equations. [Examples: the sphere defined by the equation x*x+y*y+z*z = 1 and the elliptic curve defined by y*y = x*x*x-x.] Non-commutative rings are a more diverse class of rings and hence do not allow as general a theory. There are many special classes studied. One of the motivations for the theory of non-commutative rings is the study of linear operators (a generalization of matrices -- remember, AB = BA does not usually hold for matrices). An example of a non-commutative situation occurs in quantum mechanics, where one cannot measure both the position and momentum of a particle simultaneously; whichever measurement you make first will change the other measurement. This is reflected in a mathematical way in some realizations of quantum mechanics with variables x,y such that xy and yx are different and in fact, xy-yx = 1. The Noetherian condition, however, remains an important one for non-commutative rings. Of course, ring theory is but a part of modern (abstract) algebra, and it is worthwhile to learn about modern algebra generally before specializing just in ring theory. There are many books available on ring theory and modern algebra. A couple of examples include books by R. Allenby and by J. Beachy & W. Blair, which are at a moderate level (you must have some but not much mathematical sophistication to read them). There are more advanced books (very nice books) by M. Artin, by I. Herstein, and by P. Cohn. This is a small sampling of the many, many books available.