. Let φ : R → S be a ring homomorphism of R onto S. Prove...

. Let φ : R → S be a ring homomorphism of R onto S.

Prove the following:

J ⊂ S is an ideal of S if and only if φ ^−1 (J) is an ideal of R.

Expert Solution

12345

Colby Messinger answered 2 years ago

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring...

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R. a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R). ii). Show...

Let R be a ring with at least two elements. Prove that M2×2(R)is always a ring...

Let R be a ring with at least two elements. Prove that M2×2(R)is always a ring (with addition and multiplication of matrices deﬁned as usual).

Letφ:G→G′be a group homomorphism. Prove that Ker(φ) is a normal subgroup of G. Prove both that...

Letφ:G→G′be a group homomorphism. Prove that Ker(φ) is a normal subgroup of G. Prove both that it is a subgroup AND that it is normal.

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that...

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that the symmetry group of G(S) is isomorphic to the symmetry group of S. Hint: If F is a symmetry of S, what is the corresponding symmetry of G(S)?

Consider the ring homomorphism ϕ : Z[x] →R defined by ϕ(x) = √5. Let I =...

Consider the ring homomorphism ϕ : Z[x] →R defined by ϕ(x) = √5. Let I = {f ∈Z[x]|ϕ(f) = 0}. First prove that I is an ideal in Z[x]. Then find g ∈ Z[x] such that I = (g). [You do not need to prove the last equality.]

Let φ : G1 → G2 be a group homomorphism. (abstract algebra) (a) Suppose H is...

Let φ : G1 → G2 be a group homomorphism. (abstract algebra) (a) Suppose H is a subgroup of G1. Define φ(H) = {φ(h) | h ∈ H}. Prove that φ(H) is a subgroup of G2. (b) Let ker(φ) = {g ∈ G1 | φ(g) = e2}. Prove that ker(φ) is a subgroup of G1. (c) Prove that φ is a group isomorphism if and only if ker(φ) = {e1} and φ(G1) = G2.

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.

Let F be a field and let φ : F → F be a ring isomorphism....

Let F be a field and let φ : F → F be a ring isomorphism. Define Fix φ to be Fix φ = {a ∈ F | φ(a) = a}. That is, Fix φ is the set of all elements of F that are fixed under φ. Prove that Fix φ is a field. (b) Define φ : C → C by φ(a + bi) = a − bi. Take for granted that φ is a ring isomorphism (we...

Let f : R → S and g : S → T be ring homomorphisms. (a)...

Let f : R → S and g : S → T be ring homomorphisms. (a) Prove that g ◦ f : R → T is also a ring homomorphism. (b) If f and g are isomorphisms, prove that g ◦ f is also an isomorphism.

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring...

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring of polynomials with real coefficients. For the purposes of this exercise, extend the definition of degree by deg(0) = −1. The statement to be proved is: Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there exist unique polynomials q(x) and r(x) such that f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)). Fix general f (x) and g(x). (a) Let...

Question