  Polynomial quotient rings are fundamental objects in commutative algebra. They play a big role in many areas of study, ranging from the applied sciences (cryptography, coding theory) to higher-level mathematics (algebraic geometry, field theory). Given a polynomial ring $R$ and some ideal $I \subseteq R$, a very natural question is to ask how large is the vector space dimension, or length of the quotient ring $R/I$. Specifically, are there closed-form expressions for the length of $R/I$? How does the length grow with respect to $I$? Studying the lengths of quotient rings provides insight into how polynomials with certain structures generally behave in terms of factorization, solutions to zeros, and more. Algebraists have answered this question when the ideal $I$ is of a particular form: when $I$ is a monomial ideal. However, the question becomes much more difficult and complex when $I$ becomes more general. In this project, we investigate a polynomial quotient ring in 3 variables generated by an ideal of more general form than the monomial ideal: $R/I = \mathbb{Z}_2[x, y, z]/(x^{d_1}, y^{d_2}, z^{d_3}, x + y + z)$, where $d_1, d_2, d_3$ are arbitrary nonnegative integer parameters. 
On Lengths of Quotient Rings