For each of the following ideal I in the ring Z times Z a Te

For each of the following ideal I in the ring Z times Z (a) Tell if the ideal is prime. (b) Tell if the ideal is maximal, (c) Give a ring R that is isomorphic to (Z times Z)/I. (i){(7a. 196) | a. 6 Z}. (ii){(7a,0) | a Z). (iii){(7a, 6) | a,6 Z}. (iv){(a.0) | a Z}.

Solution

1. ( 7a,19b) is not a prime. for a=2 & b=2 this is even. but a prime for a=b=1.

this ideal is maximal.

2. it is also not a prime. for a=2,3,4... because 0 is not a prime.

3. this ideal is also not a prime. for a==2 & b=2. but a prime for a=1 & b=2.

4.this ideal is also not a prime. 0 is not a prime.