(1) Prove that if R is a PID and P is a prime ideal, then R/P is another PID Show that if I is an...
I1. If p is a prime, et R be the subring The ideal I- where 1, is the ideal of z, generated by peZ, is a nil ideal of R that is not nilpotent. I1. If p is a prime, et R be the subring The ideal I- where 1, is the ideal of z, generated by peZ, is a nil ideal of R that is not nilpotent.
5.104. Let p and q be irreducible elements of a PID R. Prove that R/(pq) = R/(p) x R/(q) if and only if p and q are not associates.
Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].
3. Consider the ring R- Zz[x] and the ideal Ixx+1>, (a) Is I a prime ideal? Is I a maximal ideal? (b) Find all the multiplicative units of R/I (a+ bx cx2 a, b, c E 2. Is the group of units cyclic? If so, give a generator. If not, determine to what commor group it is isomorphic?
Please explain steps taken and why 21.21 Let R be a PID and I an ideal of R. a) Show that every ideal of R/I is principal. Must R/I be a PID? b) Show that R/I has only finitely many ideals if I is nontrivial.
Let R be a commutative ring with unity 1 and let I be a minimal ideal in R i.e. a nonzero ideal which does not properly contain another non-zero ideal. Show that either the product of two elements in I is always zero or there is an element in I that serves as unity in the ring I. Show also that in the latter case I is a field.
11.rpis a prime, let R be the subring ΣΖ,n of 11 Zpn. The ideal 1- where I, is the ideal of Z generated by p eZ, is a nil ideal of R that is n nilpotent. ot 11.rpis a prime, let R be the subring ΣΖ,n of 11 Zpn. The ideal 1- where I, is the ideal of Z generated by p eZ, is a nil ideal of R that is n nilpotent. ot
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine how many elements Zu/5+5i) has. (5) Let m,n be integers with m|n. Prove that the surjective ring homomor- phism Z/n -> Z/m induces a group homomorphism on units, and that this group homomorphism is also surjective. (4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine...
10 Let R be a commutative domain, and let I be a prime ideal of R. (i) Show that S defined as R I (the complement of I in R) is multiplicatively closed. (ii) By (i), we can construct the ring Ri = S-1R, as in the course. Let D = R/I. Show that the ideal of R1 generated by 1, that is, I R1, is maximal, and RI/I R is isomorphic to the field of fractions of D. (Hint:...
Let a and b be non-zero elements of a principal ideal domain R, and let 1 = (a) and I = (6). Show that the following are cquivalent: (i) I and I are comaximal. (ii) In J = II. (iii) ab is a least common multiple of a and b. (iv) 1 = ged(a,b).