I am trying to show that a ring with 48 elements is not an integral domain.
Let $R$ be a ring with 48 elements. I know I need to show that $ab = 0$ for some nonzero elements $a , b \in R$ in order to conclude that $R$ cannot be an integral domain. But I'm not seeing how to use the fact that the ring has 48 elements to make progress towards this.
Am I supposed to identity $R$ with some other ring 48-element ring that I can actually make algebraic calculations with? That would be a big help. Otherwise, I don't know what the elements of $R$ are, and so I can't begin to try to find the appropriate elements $a, b \in R$.
I don't know of any results that can help me identity a 48-element ring $R$ with another ring. I only know of such results with fields (the classification of finite fields, for example.)