Question: There is a question having several parts. I solved the part (a) but I have problems with other parts. 

Let 
$f\in L^{1}(\mathbb{R}^{n}),$
with
$f\geq 0\;$ a.e. 
in 
$\mathbb{R}^{n}$
and let 
$g:\mathbb{R}^{n}\to [0,+\infty)$
be the function defined by 
$$ g(x):=\int\_{B(x,\vert x\vert/2)}f(y)dy,$$
where 
$\vert \cdot\vert$ 
denotes the Euclidean norm and 
$B(x,r)=\{y\in \mathbb{R}^{n}:\vert y-x\vert <r\}$
for every 
$x\in \mathbb{R}^{n}$
and every
$r\geq 0.$

(a) Prove that 
$g$ 
is continuous. 

(b) Prove that $g(x)\to 0$
as
$\vert x\vert\to \infty. $

(c) Prove that 
$g$
has a maximum point in 
$\mathbb{R}^{n}.$

Please write only hint and not a whole solution, I have to be accustomed to solve the mathematics problems. A small useful hint will be accepted. Also, if you have seen this problem in other books, please let me know.

Answer: 
Note that the zero element of $\mathbb{Z}\oplus \mathbb{Z} / I $ is $I=0+I.$

Let $$((x\_1,y\_1)+I)((x\_2,y\_2)+I)=0+I$$ 
Then $$(x\_1x\_2,y\_1y\_2)\in I$$
Thus $$7\mid y\_1y\_2$$ 
As $7$ is prime, therefore $7 \mid y\_1$ or $7 \mid y\_2$. Hence, $(x\_1,y\_1) \in I$ or $(x\_2,y\_2)\in I$. 

So, $(x\_1,y\_1)+I = 0 + I$ or $(x\_2,y\_2)+I=0+I$. 
So, $\mathbb{Z}\oplus \mathbb{Z} / I $ is an integral domain.
It would be a good exercise to prove the following results.

> 
> **Result-1:** In $\Bbb Z \times \Bbb Z$ the prime ideals are of the form, $\Bbb Z/p\Bbb Z\times \Bbb Z$ or $\Bbb Z\times\Bbb Z/p\Bbb Z $ where $p$ is prime.
> 
> 
> 

In fact you can generalize the above result as follows.

> 
> **Result-2:** Let $R\_i$ be a comutative ring with unity for each $i=1,...,n$. The prime ideals of $R\_1\times ...\times R\_n$ are of the form $P\_1\times...\times P\_n$ where for some $j$, $P\_j$ is a prime ideal of $R\_j$ and $ P\_i=R\_i$ for $i\not=j$.
> 
> 
> 

In your case, $I=\Bbb Z \times \Bbb Z/7\Bbb Z$. Hence it follows that $I$ is a prime ideal.