TITLE: Do people study "ring presentations"? Is this a dumb question?
QUESTION [11 upvotes]: So one way to define a group presentation is to say, well, let's generate the free group with some number of generators, and then quotient by saying certain elements (relators) cancel just as $aa^{-1}$ does (and other things, you have to take the normal subgroup generated by the elements, but the basic idea works, I think). That is, we have for some words $R_1, R_2, \ldots$ the relations $R_1 = R_2 = \ldots = e$.
A simple example is the cyclic group of order n: one presentation is $\langle g \mid g^n \rangle$.
My question is: can and do we do this for rings? I'm imagining you would do something like take the "free ring" consisting of all sums of products of generators, and perhaps instead of having all the relators equal to the trivial group they would equal the zero ideal, because then if you wanted some word $R$ to equal the multiplicative identity you would just say $R - 1 = 0$. (In particular, the comments on the vaguely related question that got me thinking about this is here: could you formulate a rigorous proof by "dividing by the relations"?)

REPLY [2 votes]: You can consider presentations for any algebraic system given by special constants, operations and equations.  For example, groups are given by the special constant 1 (for the multiplicative identity), the unary operation of inverse, x-1, and the binary operation of multiplication, $xy$, the plus the equations that these must satisfy.  In that case, you can specify every object with such operations in terms of a set of generators, and the relations that the generators must satisfy.  The study of these kinds of constructions is known as universal algebra.
For the specific case of commutative rings over a field, there is an explicit algorithm for working with presentations, known as the Groebner basis algorithm.  It's widely implemented in symbolic algebra packages.  (In the case of commutative rings, you're basically just working with systems of polynomials.)