(4) Let R be an integral domain and o n,b,ce R such that ab = ne....
2. Let R be an integral domain containing a field K as a unital subring. (a) Prove that R is a K-vector space (using addition and multiplication in R). (b) Let a be a nonzero element of R. Show that the map is an injective K-linear transformation and is an isomorphism if and only if is invertible as an element of R. (c) Suppose that R is finite dimensional as a K-vector space. Prove that R is a field.
First: As I mentioned in my e-mail, a Euclidean valuation on an integral domain R is a function u : R* → N (where R* is the set of nonzero elements of R, and N includes 0) with two properties: (1) if a,b E R*, thern (a) v(ab); and (2) if a, b R and b 0, then there exist elements q,r R such that a-bqr and either 0 or v(r) < v(b). Prove that if o is a Euclidean...
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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.
Let R be an ED but not a field, with a norm function N R-Z U0 such that N(ab for all a, bE R. (a) Prove that N(1R) (b) If r E R is a unit, show that N(r) 1. (c) If r E R is nonzero, show that N(r) 0. (d) For any r E R, prove that if N(r) 1, then r is a unit N(a)N(b) e) For any r e R if N() is a prime mumber,...
8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain for which a2a 1-0 but a f -1.
8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain...
Suppose (Z, +, ·) is an ordered integral domain. Let a and b and c be elements of Z such that c^2 + a · c + b = 0. (a) Prove that if 4b = a^2 , then x^2 + a · x + b ≥ 0 for all x ∈ Z. (b) Prove that if 4b =/= a^2 , then there is exactly one element d in Z such that d^2 + a · d + b =...
1. Let H- ta + bija, b e R, ab 20). Prove or disaprove that H is a subgroup of C under addition. 2. Let a and b be elements of an Abelian group and let n be any integer. Prove that (ab)"- a
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0 4. Let A and B be n x n such that B 1-A and A2 = A. Show that AB-BA-0
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0
4. Let A and B be n x n such that B...
Exercise 5.3.4: Let f: [a,b] → R be a continuous function. Let ce [a,b] be arbitrary. Define po the Prove that F is differentiable and that F'(x) = f(x) for all x € [a,b]. series on the
Let R={1 € Q[2] : [0) € Z}. (a) Show that R is an integral domain and R* = {+1}. (b) Show that irreducibles of Rare Ep for primes pe Z, and S() ER with (0 €{+1} which are irreducible in Q[r]. (c) Show that r is not a product of irreducibles, and hence R does not satisfy the ascending chain condition for principal ideals.