Let R={1 € Q[2] : [0) € Z}. (a) Show that R is an integral domain...
From Goodman's "Algebra: Abstract and Concrete" 6.6.7. Show that R = Z + xQ[x] does not satisfy the ascending chain condition for principal ideals. Show that irreducibles in R are prime.
Suppose that R is a domain satisfying the ascending chain condition on principal ideals. Show that R is a UFD if and only if every pair of elements have a greatest common divisor. Suppose that R is a domain satisfying the ascending chain condition on principal ideals. Show that R is a UFD if and only if every pair of elements have a greatest common divisor.
Let a EC Z such that a? EZ and Rea=0. Let N: Z[0] → Z:2H |212. (a) Show that N() NU {0} for all ze Z[a], and that if ry in Za], then N (2) N(y). (b) Show that Z[a] satisfies the ascending chain condition for principal ideals. (e) (Bonus) Show that Z[iV2 is a Euclidean domain. (Hint: there's a proof in the textbook that Z[i] is a Euclidean domain that you can modify.) (d) Show that Ziv5) is not...
Let a EC Z such that a? EZ and Rea=0. Let N: Z[0] → Z:2H |212. (a) Show that N() NU {0} for all ze Z[a], and that if ry in Za], then N (2) N(y). (b) Show that Z[a] satisfies the ascending chain condition for principal ideals. (e) (Bonus) Show that Z[iV2 is a Euclidean domain. (Hint: there's a proof in the textbook that Z[i] is a Euclidean domain that you can modify.) (d) Show that Ziv5) is not...
Let a EC Z such that a? EZ and Rea=0. Let N: Z[0] → Z:2H |212. (a) Show that N() NU {0} for all ze Z[a], and that if ry in Za], then N (2) N(y). (b) Show that Z[a] satisfies the ascending chain condition for principal ideals. (e) (Bonus) Show that Z[iV2 is a Euclidean domain. (Hint: there's a proof in the textbook that Z[i] is a Euclidean domain that you can modify.) (d) Show that Ziv5) is not...
Let R be a UFD, and let So be a set of irreducibles in R. Let S := {ufi.fr: k > 0,[1,...,SE E So, u € R*} (we use the convention that the product fifk is 1 when k=0). (a) Show that S is multiplicatively closed. (b) Suppose / ER, GES. Show that is a unit in S-R if and only if SES. (c) Show that res-'R is irreducible if and only if x is associates with y = {es-R,...
4[10 pts]. Let f(z) = u (r,0) + iv(r,0) be analytic in a domain D c C which does not contain the origin. Then do the following ones: (a) Show that rurr(r, θ) + rur(r, θ) + u69(r, θ) 0 for all re® E D. (b) Show that (a) is equivalent to the condition that u is harmonic in D (c) Show that the function (in|e )2-[Arg( a(z) z)]2,-π < Arg(z) < π, 4[10 pts]. Let f(z) = u (r,0)...
37. Show that if D is an integral domain, then 0 is the only nilpotent element in D. 38. Let a be a nilpotent element in a commutative ring R with unity. Show that (a) a = 0 or a is a zero divisor.. (b) ax is nilpotent for all x ER. (c) 1 + a is a unit in R. (d) If u is a unit in R, then u + a is also a unit in R.
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...
Please help! Thank you so much!!! 2. (10 points) Let R be an integral domain and M a free R-module. Prove that if rm 0 or m 0 where r E R and m E M, then either r 0. 2. (10 points) Let R be an integral domain and M a free R-module. Prove that if rm 0 or m 0 where r E R and m E M, then either r 0.