Prove the following: (a) Let A be a ring and B be a field. Let f...

Prove the following:

(a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal.

(b) If A/J is a field, then J is a maximal ideal.

Expert Solution

Colby Messinger answered 2 years ago

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...

A theorem for you to prove: Let B be a local ring containing a perfect field...

A theorem for you to prove: Let B be a local ring containing a perfect field k that is isomorphic to its residue field B/m, and such that B is a localization of a finitely generated k-algebra. Then the module of relative differential forms M_B/k is a free B-module of rank equal to the dimension of B if and only if B is a regular local ring.

2 Let F be a field and let R = F[x, y] be the ring of...

2 Let F be a field and let R = F[x, y] be the ring of polynomials in two variables with coefficients in F. (a) Prove that ev(0,0) : F[x, y] → F p(x, y) → p(0, 0) is a surjective ring homomorphism. (b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x, y) + ys(x, y) | r,s ∈ F[x, y]} (c) Use the first isomorphism theorem to prove that (x, y) ⊆ F[x, y]...

Let F be a field, and recall the notion of the characteristic of a ring; the...

Let F be a field, and recall the notion of the characteristic of a ring; the characteristic of a field is either 0 or a prime integer. Show that F has characteristic 0 if and only if it contains a copy of rationals and then F has characteristic p if and only if it contains a copy of the field Z/pZ. Show that (in both cases) this determines the smallest subfield of F.

Question

Prove the following: (a) Let A be a ring and B be a field. Let f...

Solutions

Expert Solution

Related Solutions

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

A theorem for you to prove: Let B be a local ring containing a perfect field...

2 Let F be a field and let R = F[x, y] be the ring of...

Let F be a field, and recall the notion of the characteristic of a ring; the...

Let A be a commutative ring and F a field. Show that A is an algebra...

4. Let a < b and f be monotone on [a, b]. Prove that f is...

Prove the theorem in the lecture:Euclidean Domains and UFD's Let F be a field, and let...

Let F be a field and R = Mn(F) the ring of n×n matrices with entires...

Let F be a finite field. Prove that the multiplicative group F*,x) is cyclic.

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