Let R be the ring of all 2 x 2 real matrices. A. Assume that A...

Let R be the ring of all 2 x 2 real matrices.

A. Assume that A is an element of R such that AB=BA for all B elements of R. Prove that A is a scalar multiple of the identity matrix.

B. Prove that {0} and R are the only two ideals.

Hint: Use the Matrices E11, E12, E21, E22.

Solutions

Expert Solution

milcah answered 1 year ago

Next > < Previous

Related Solutions

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring...

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring of the functions from X to R. Show directly the associativity of the multiplication of F(X, R). Assume that R is unital and commutative. show that F(X, R) is also unital and commutative.

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 V = R^2×2 be the vector space of 2-by-2 matrices with real entries over the...

Let V = R^2×2 be the vector space of 2-by-2 matrices with real entries over the scalar field R. We can define a function L on V by L : V is sent to V L = A maps to A^T , so that L is the “transpose operator.” The inner product of two matrices B in R^n×n and C in R^n×n is usually defined to be <B,C> := trace (BC^T) , and we will use this as our inner...

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).

Let V be the vector space of 2 × 2 real matrices and let P2 be...

Let V be the vector space of 2 × 2 real matrices and let P2 be the vector space of polynomials of degree less than or equal to 2. Write down a linear transformation T : V ? P2 with rank 2. You do not need to prove that the function you write down is a linear transformation, but you may want to check this yourself. You do, however, need to prove that your transformation has rank 2.

Find all of the ideals of Q, M2(R) (the 2 x 2 matrices with entries in...

Find all of the ideals of Q, M2(R) (the 2 x 2 matrices with entries in R) and M2(Z) (the 2 x 2 matrices with entries in Z) and determine which ideals are maximal and which ideals are prime. Please explain why the ideals are maximal and/or prime.

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.

c) Let R be any ring and let ??(?) be the set of all n by...

c) Let R be any ring and let ??(?) be the set of all n by n matrices. Show that ??(?) is a ring with identity under standard rules for adding and multiplying matrices. Under what conditions is ??(?) commutative?

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).

Subjects