Do people study "ring presentations"? Is this a dumb question?
Solution 1:
You certainly can: the constructions of the tensor algebra, symmetric algebra, and exterior algebra can all be performed using generators and relations (though are perhaps usually thought of conceptually via a universal property). The operation of localization is also an example of this: one formally adjoins "denominators" satisfying the requisite relations (localization of $A$ at a multiplicative set $S$ can be thought of adjoining variables $x_s, s \in S$ with the relations $s x_s = 1$).
The notion of a ring having a "finite presentation" over another ring $R$ (i.e. be the quotient of a polynomial ring $R$ by a finitely generated ideal--that is, by finitely many generators and relations) is of importance in algebraic geometry, as many properties proved in, say, the finite type case over noetherian schemes extend here (e.g. I believe Chevalley's theorem of the constructibility of the image of a constructible set under a morphism of schemes is true under finite presentation hypotheses). I am not sufficiently qualified to say much more on this topic, but you might find BCnrd's answer here enlightening.
Solution 2:
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.)