I am trying to solve this question:
Show that the ring M2(F2) of 2×2 matrices with entries from F2 is non-commutative. Here F2 is the field of 2 elements
Using a counterexample I can easily show that the ring M2(R) is non-commutative. I assume this question can be done in a similar way. However I am unsure on what the elements of the 2x2 matrix should be. In other words, what is the field of 2 elements?
On
The field of 2 elements is $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z},$ i.e. it's a residue field. Equipped with the operations $+$ and $\cdot$, it only has the additive unit 0 and the multiplicative unit 1 as elements.
It's the field consisting only of $0$ and $1$, with the operations $+$ and $\cdot$ modulo $2$, so $1+1=0$ in this field.