Question

In: Advanced Math

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

Let F be a field and R = Mn(F) the ring of n×n matrices with entires in F. Prove that R has no two sided ideals except (0) and (1).

Expert Solution

Let be a two sided ideal in . We show that where denotes the identity matrix. Since , there exists a nonzero matrix . Let . Then, by applying some elementary row and column operations on we can transform it to

where is the identity matrix.

Since the elementary row operations are applied just by multiplying by an elementary matrix on left and a column operation by multiplying by an elementary operation on right, we get there exist elementary matrices such that

and this shows that

since is a two sided ideal in .

Now just by multiplying

by permutation matrices on left and then on right we get that contains several matrices having and on diagonal and all other entries and by adding some of them we get that contains a digonal matrix having all diagonal entries nonzero.

Now, if we just multiply that diagonal matrix by an appropriate diagonal matrix, we get that contains the identity matrix and hence it contains al the matrices from which shows that

Colby Messinger answered 1 year 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...

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 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 R be a ring and f : M −→ N a morphism of left R-modules. Show that:

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that: c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels. d) N/f(M) satisfies the Universal Property of Cokernels. Q2. Show that ZQ :a) contains no minimal Z-submodule

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that: c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels. d) N/f(M) satisfies the Universal Property of Cokernels.

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the...

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the characteristic polynomial of T. You may wish to use the basis of Mn×n(F) consisting of the matrices eij + eji, eij −eji and eii. 2. Let A = (a b c d) (2 by 2 matrix) and let T : M2×2(F) → M2×2(F) be deﬁned asT (B) = AB. Represent T as a 4×4 matrix using the ordered basis {e11,e21,e12,e22}, and use this matrix to...

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.

Let A∈Mn(R)"> A ∈ M n ( R ) A∈Mn(R) such that I+A"> I + A I+A is invertible. Suppose that

Let A∈Mn(R)A∈Mn(R) such that I+AI+A is invertible. Suppose thatB=(I−A)(I+A)−1B=(I−A)(I+A)−1(a) Show that B=(I+A)−1(I−A)B=(I+A)−1(I−A) (b) Show that I+BI+B is invertible and express AA in terms of BB.

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.

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

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

Question

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

Solutions

Expert Solution

Related Solutions

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

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

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

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the...

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

Let A∈Mn(R)"> A ∈ M n ( R ) A∈Mn(R) such that I+A"> I + A I+A is invertible. Suppose that

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...