Let $$F:=\left \{ \begin{pmatrix} a & b\\ 2b & a \end{pmatrix}\left. \right | a,b \in \mathbb{F}_5 \right \}$$ Show that: $F$ is a ring with addition and multiplication given by matrix multiplication in $Mat(2;\mathbb{F_5})$.
This question seems relatively simple, but I'm quite new to fields and rings and there are some things regarding notation and terminology that I'm unsure about.
In the equation for $F$, the $:=$ symbol is used, would I be correct to assume that it has no real special meaning and is equivalent to the $=$ symbol?
Since all fields are rings, it seems strange that the questions would ask to show that $F$ is a ring.
As far as I understand, $Mat(2;\mathbb{F}_5)$ is the map of all $2 \times 2$ matrices to the field $\mathbb{F}_5$, so is the question asking me to take $x,y,z \in F$ and show that they satisfy the ring axioms? This is the part where I'm really struggling to understand what the question is asking.
For the first part, typically the := means “is defined as”. That notation just represents that it is being defined this way. For the second part, are you assuming that it is a field because it is labeled F? You would definitely need to show it’s a ring before it’s a field which I’m sure that you know. For the third part, I interpret $M(2;\mathbb{F}_5)$ as the space of $2\times2$ matrices with coefficients in the finite field with 5 elements. Addition and Multiplication are defined by matrix addition and multiplication but you need to be careful since you’re working in $\mathbb{Z}/5\mathbb{Z}$. So really the last part is just asking you to show that matrices of the given form form an abelian group with standard matrix addition and satisfy the ring multiplication axioms with standard matrix multiplication.