Question related to Ideals of the ring of all functions from the set $\{1,2, \dots , 10\}$ to $\mathbb Z_2 $.

Question

I have the following question in a competitive exam , but I failed to answer it.The question is:

Let $\mathcal R = \{f:\{1,2, \dots , 10\} \rightarrow \mathbb Z_2\} $ be the set of all $\mathbb Z_2$-valued functions on the set $\{1,2,\dots , 10\}$. Then $\mathcal R$ is a commutative ring with pointwise addition and multiplication of functions.Which of the following statements are correct?

1. $\mathcal R$ has a unique maximal ideal.
2. Every prime ideal of $\mathcal R$ is also maximal.
3. Number of proper ideals of $\mathcal R$ is 511.
4. Every element of $\mathcal R $ is idempotent.

The only option I was able to answer was option 4 .

Can anyone help me understanding the ring in the question and the options.

Any insight will be happily appreciated. Thank you.

Answer 1

Hint for 1: The ideal consisting of functions $f$ such that $f(3)=0$ is maximal; is $3$ special?

Hint for 2: What's a finite domain?

Hint for 3: I bet on 1023, because $2^{10}=1024$.

Statement 4 is indeed true, because every element in the two element ring is idempotent.