Ideals in the ring of Gaussian integers 
*

*What are the proper ideals, prime ideals, maximal ideals of $\mathbb{Z}[i]$, the ring of Gaussian integers.   

*Check whether $(1+i)$ is prime or maximal ideal.



Can someone help me please. I have no idea how to crack this problem. Thanks for your time.

$\mathbb Z[i]$ is a Euclidean domain, and thus a principal ideal domain. So every ideal is generated by a single element, this should help you give a description of the proper ideals. The prime ideals are related then to the prime elements which are known as Gaussian primes. Have a look at this for more information. 
