Let R and S be rings. If R is isomorphic to S, show that R[x] is...

Let R and S be rings. If R is isomorphic to S, show that R[x] is isomorphic to S[x].

Expert Solution

Colby Messinger answered 8 months ago

Let R and S be rings. Denote the operations in R as +R and ·R and...

Let R and S be rings. Denote the operations in R as +R and ·R and the operations in S as +S and ·S (i) Prove that the cartesian product R × S is a ring, under componentwise addition and multiplication. (ii) Prove that R × S is a ring with identity if and only if R and S are both rings with identity. (iii) Prove that R × S is a commutative ring if and only if R and...

Let R and S be commutative rings with unity. (a) Let I be an ideal of...

Let R and S be commutative rings with unity. (a) Let I be an ideal of R and let J be an ideal of S. Prove that I × J = {(a, b) | a ∈ I, b ∈ J} is an ideal of R × S. (b) (Harder!) Let L be any ideal of R × S. Prove that there exists an ideal I of R and an ideal J of S such that L = I × J.

Let G = R\{0} and N = (0,∞). Show that G/N is isomorphic to the multiplicative...

Let G = R\{0} and N = (0,∞). Show that G/N is isomorphic to the multiplicative group {1, −1}.

6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic to the...

6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic to the subfield of C consisting of complex numbers with rational real and imaginary parts.

Let A = R x R, and let a relation S be defined as: “(x1, y1)...

Let A = R x R, and let a relation S be defined as: “(x1, y1) S (x2, y2) ⬄ points (x1, y1) and (x2, y2)are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you must give a counterexample.

Let L = {x = a^rb^s | r + s = 1mod2}, I.e, r + s...

Let L = {x = a^rb^s | r + s = 1mod2}, I.e, r + s is an odd number. Is L regular or not? Give a proof that your answer is correct.

Let L = {x = a r b s c t | r + s =...

Let L = {x = a r b s c t | r + s = t, r, s, t ≥ 0}. Give the simplest proof you can that L is not regular using the pumping lemma.

Let R and S be nontrivial rings (i.e., containing more than just the 0 element), and...

Let R and S be nontrivial rings (i.e., containing more than just the 0 element), and define the projection homomorphism pi_1: R X S --> R by pi_1(x,y)=x. (a) Prove that pi_1 is a surjective homomorphism of rings. (b) Prove that pi_1 is not injective (c) Prove that (R X S)/R (not congruent) R. Please show all parts of answer

5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f...

5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f is injective, but not surjective. (b) Suppose g : R\{−1} → R\{a} is a function such that g(x) = x−1, where a ∈ R. Determine x+1 a, show that g is bijective and determine its inverse function.

The Chinese Remainder Theorem for Rings. Let R be a ring and I and J be...

The Chinese Remainder Theorem for Rings. Let R be a ring and I and J be ideals in R such that I + J = R. (a) Show that for any r and s in R, the system of equations x ≡ r (mod I) x ≡ s (mod J) has a solution. (b) In addition, prove that any two solutions of the system are congruent modulo I ∩J. (c) Let I and J be ideals in a ring R...

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