I'm trying to understand how we can systematically study the factor rings of polynomials over a ring K. For example imagine that we're working in $R=K[x_1,...,x_n]$ and we have the ideal $I=(p_1(x_1,...,x_n),...p_m(x_1,...,x_n))$.
1- Is it always possible to find the multiplication rule in the quotient ring $R/I$ using Gröbner bases?
2- Do we really need Gröbner bases for that? I mean I can happily work in $\mathbb{C} \cong \mathbb{R}/(x^2+1)$ without knowing anything about Gröbner bases, I can also work in Tessarines and Quaternions with confidence without bothering about the associated factor rings.
3- Is it the purpose of studying Gröbner bases or we can study factor rings of $K[x_1,..,x_n]$ without them? I mean Gröbner bases were invented because we wanted to study the ideals of the polynomial rings over a finite number of variables and their factor rings or there's more than that about Gröbner bases?
Groebner basis are mainly a computational tool rather than a theoretical one. For example, they are important in solving zero-dimensional systems of polynomial equations or for eliminating redundant variables in underdetermined polynomial systems of equations. Consequently, you can very well study quotients of polynomial rings without any knowledge of Groebner basis. Conversely, many techniques of Groebner basis apply to quotients of polynomial rings, i.e. to finitely-generated algebras over a field or a ring.