Hello,
Kindly Can you help me in answer this problem:
Let Z[X] be the ring of polynomilas of integers
i) Show that J= 2Z[X]+xZ[X] is not principal
ii) Show that I=4Z[X]+xZ[X] is a primary ideal of Z[X]
For (i), recognize J as the set of polynomials with even constant term. Any principal generator has to divide everything in J, which in particular contains all the constant polynomials {2, 4, 6, ...}. The only possibilities are 1 and 2. Since J is not the whole ring, 1 is not a generator. And 2 doesn't generate all of J, since it generates polynomials with all even coefficients.
For (ii), a hint: describe I similarly to how we described J above. If pq is in I, then what do you know about the constant terms of p and q?
Thaaaaaank you very much for your help I understood well the first question and regarding the second I think that we can describe I as the set of all polynomials in z[x] with multiple of 4 constant term. In other word I is contained in J above. Then if pq is in I, that's means the constant term of pq is multiple of 2, but the constant term is actually the product of constant terms of p and q. So if 2 does not divide the constant term of p, it must devide the constant term of q.
Thank you again and if there is any mistake or comments, please guide me