We were told in class that a $K$-basis for $S/I$ where $S=K[X_1, \dots , X_n]$ and $I$ a monomial ideal in $S$ is $W = \{X^a \in \mathrm{Mon}(S) \mid X^a \notin I\}$.
I'm having difficulties visualizing what the elements of the basis would be like and would appreciate if someone can explain the abstract definition with a minimal example such as $S=K[X_1, X_2]$ and $I=\langle X_1^a, X_2^b \rangle (a, b \ge 1)$ and write explicitly what sort of elements would be in the basis of the quotient ring. Thanks for your help.
A basis for $S/I$ over $K$ when $a=2,b=3$ is made of the (images) of the following monomials: $1,X,Y,XY,Y^2,XY^2.$ (Think about all the monomials of degree $d$, for $d=0,1,\dots$ and check which one are in $I$. In this particular case, for $d\ge4$ all of them are in $I$.)
In general, a monomial $m\in I$ iff it is divisible by a monomial generator of $I$. This tells us that in order to find out a $K$-basis of $S/I$ we have to remove all such monomials.