Answered: Let R be a ring and let M be a free…

Let R be a ring and let M be a free R-module with basis B. If N is any R-module and f: B→ N is any mapping, then there exists a unique R-module homo- morphism : M→N such that 18 = f.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: 5. Let F = Q(√2, √3) be the smallest subfield of C that contains √2 and √3. (a) Find a basis for F…

A: Given : F = ℚ2, 3 be the smallest subfield of ℂ that contains 2 and 3 To find : (a) The basis for F…

Q: . Let R be the ring of all complex numbers and R' the a b b a ring of all 2 x 2 matrices of the form…

Q: Let M be a finitely generated free module over a commutative ring. Then all basis of M are finite.

A: Given is that M is be a finitely generated free module over a commutative ring then we have to show…

Q: Let R be a ring in which each element is idempotent. Let R = R XZ₂. Define + and x on R by (a, [n])…

A: Given: R is a ring in which each element is idempotent and R=R×Z2.To show: + and × are well-defined…

Q: Let R be a regular ring with 1. (i) Prove that for any a E R, there exists an idempotent e ER such…

A: As per the question, we consider regular rings with unity. We prove two results related to…

Q: If Pn is the space of polynomials of degree at most n with a basis B = {1, x, x², ..., xª}, compute…

Q: A) Find MS (B,C) B) Find a basis for the kernel of S C) Find a basis for the range of S D)…

Q: Find every possible ring homomorphism from Z15 to Z25 . For each one, compute the kernel and image.

A: To find all the possible ring homomorphisms from Z15 to Z25, we need to consider where the generator…

Q: %3D

Q: Let R be a ring with unity which has exactly one maximal left ideal M. Show that the only…

A: Given: R is a ring with unity which has exactly one maximal left ideal M. To show: The only…

Q: 3. Find a basis for the vector space RX] of all polynomials in X having degree at most 4 such that…

A: We have to find a basis for the vector space R4[x] of all polynomials in X having degree atmost 4…

Q: Let R be a ring with identity, f: RS be an onto homomorphism and r ER be a unit. Show that f(r) ES…

Q: _C 16. Suppose R is commutative and let I and J be ideals of R, so R/I and R/J are naturally…

Q: Exercise 3: Let R be a commutative ring. (a) Let I and J be ideals in R. Show that In J is also an…

Q: Which sets are equal to the set (a, b, c)? {b, b, b, x, x, x, C, C) {b, b, b, b, b, c, c, c, a, a)…

A: What is Set: Set is the language of mathematics. Through set, we can define mathematical expressions…

Q: -se A, B are submodules (that is, they are isom : M/A→M/B and show pism, then use the FHT

Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: et J be the ideal of R = Cla, b, c, d] generated by a²b, b³c, cd, d5 a. a) Write A = V(J) as a union…

A: Given: The Given Ideal of R=Ca,b,c,d generated by a2b,b2c,c4d,d5a. To write A=VJ as a union of…

Q: If R is a system satisfying all the conditions for a ring with unit element with possible exception…

A: To prove that the axiom a+b=b+a must hold in R and R is thus a ring

Q: Let R be the ring Z[V3] = {a+b/3|a, b E Z}. For each of the following elements of R determine if it…

A: let R be the ring ℤ3=a+b3|a,b∈ℤ claim- for each element of R determine if it is an invertible…

Q: phism, where R is a commutative ring with unity. Le he arbitrary elements of R. Then there exists a…

Q: Let R be a field of real numbers and M₂(R) be a ring of 2 x 2 matrices over R under usual…

Q: if T is an operator in a real vectorial space with characteristic polynomial (x²-2x+5)². What are…

A: For the solution follow the next steps.

Q: (5) I is maximal ideal of ring R if and only if R = (a, I) for any..... a in R O a in I a in R-{I} o…

Q: {[: :]we2} b a, b E Z} and let o : R → Z be defined by ø a a Let R = a a а — b. (1) Show that ø is a…

Q: Consider H = {[x]x, y ≤ Z} and the mapping of f: H → Z given that f([x]) = x − y.

Q: Suppose R is commutative. Prove that the map oa : R[x] → R where $₁(p(x)) = p(a) for some a € R is a…

Q: Let R be a ring and let I a minimal left ideal of R. If M is a faithful simple left R-mod

A: R be a ring and I be a minimal left ideal of R

Q: Consider the set of matrices S = -{(a+b) a + b) : ab € R } Consider the function f: S→ R defined as…

Q: 4[X]- = 9x₁x₂. representation P for L relative to B₁ and B₂ such that [L(u)]₂=P[u]₁ Let L: R2 R2 be…

A: First of all we find the image of basis elements of domain and write them as a linear combination of…

Q: Let M be Noetherian left R-modu

Q: Suppose X and Y are rings and that f : X → Y is an isomorphism. Let X2x2 be the set of all 2 × 2…

Q: Let R be a ring and M a module over R. Let S be a non- empty set and let {x; : i E S} be a basis of…

A: Given that R be a ring and M a module over R. Let S be a non-empty set and let {xi : i∈ S } be a…

Q: Let R = R[x] be the ring of real polynomials in the variable x, and let I = be the ideal of R…

A: Given is the ring of real polynomial in the variable of is the ideal of generated by For integer ,…

Q: Prove That Let X and Y be two fuzzy rings of R and R' respectively and f: R R is a homo. Then 1- if…

Q: Let R be a commutative ring and M₂(R) the ring of 2-by-2 matrices with entries in R (with the usual…

A: In the given proof, we showed that if I is an ideal in a commutative ring R, then the set M2(I) of…

Q: Let M be an R-module, and let N be an R-submodule of M. Then M is noetherian (artinian) if and only…

A: Given:M is an R-module, and N is an R-submodule of M. To Prove:M is Noetherian (Artinian) if and…

Q: . Let R be the ring of all complex numbers and R' the ring of all 2 x2 matrices of the form a b b a…

Q: rove that any unital irreducible R-module is cyclic.

A: Any unital irreducible R-module is cyclic

Q: If R is a ring and S is a subring of R, then S is an R-module (module over R).

A: R is ring and S is a subring of R Claim : S is an R -module (module over R)

Q: Let f: Z6→ Z2 × Z3 be defined by the following specification of its values 0 → (0, 0) , 1 → (1, 1) ,…

Q: Let : A B be a mapping of sets, A a system of subsets of A, and 3 a system of subsets of B. Define…

A: Since you have asked multiple questions, we will solve the Question (k) and (l) for you. If you want…

Q: Let V = span {1+ 2x – x² + 3r³, -2+x+x² – x*, -1+ 8r – x² + 7r", 3 + x – 2r² + r'}. Find a basis ß…

A: Step:-1 Note:- To span V we need the polynomials of degree 3, so dimension of V is 4. (think using…

Q: Let R be a ring and M a left R-module. Show that, for any idempotent e ER End(M) We have ker(e) =…

A: Step 1: Given thatker(e)⊆im(Id−e)im⁡(Id−e)⊆ker(e) Step 2:ker(e)⊆im(Id−e)Take x in ker(e). This means…

Question

Let R be a ring and let M be a free R-module with basis B. If N is any R-module and f: B→ N is any
mapping, then there exists a unique R - module homo- morphism : M→ N such that 018 = f.

Expert Solution

Step 1: Given information

Let R be a ring and let M be a free R-module with basis B. If N is any R-module and $f colon B rightwards arrow N$ is any mapping, then we have to show that there exists a unique R-module homomorphism $ϕ colon M rightwards arrow N$ such that $empty set subscript i B end subscript equals f$ .

Step by step

Solved in 3 steps with 11 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

(Fundamental theorem an module homomorphism) Let M, M' be R-mod- ules and f: M→ M' be an R-module homomorphism. Then Ker (f) is a submodule of M and M/Ker (f) Im (f). Equivalently, every homomorphic image of an R-module is isomorphic to some quotient module.
2. Let R be the commutative ring of all matrices with rational entries, M3(Q), of the form a b c 0 ab 00 a Show that the mapping : R Q given by ab c 0 ab 00a is a homomorphism. $ - a
ring theory
Let R be a ring. Consider the map Ø:Q[x]→Q defined by Ø(f(x))=f(3). Then, the Kernel of Ø is: * O None of the choices O O
Let R be a commutative ring with unity and let S be a nonempty subset of R. Show that I(S) := {f(x) = R[x] : f(a) = 0, for all a € S} is an ideal of R[x].
Show that T is an Let T : P₁ → R² be defined as isomorphism. T (a + bt) = a+
It is not always possible to construct by ruler and compass, a square equal in area to the area of a circle.
2. Assume that the set S I, y is a ring with respect to matrix addition and multiplication. D-r is a ring homomorphism from (a) Verify that the mapping e : S Z defined by e R to Z. (b) Find ker 0. (c) Find S/kere. (d) Find an isomorphism from S/kere to Z.
Let R be the ring Z[V2] = {a + b/2|a, b € Z}. For each of the following elements of R determine if it is an invertible element of R, an irreducible element of R, or neither invertible nor irreducible: (i) V2, (ii) 1+ v2, (iii) 2 + v2, (iv) 3 + v2, (v) 4 + v2, (vi) 5 + V2. Hint: Use the multiplicativity of the function R → Z that sends m + n/2 to m² – 2n².
Let R be a commutative noetherian ring and let M be a uniform R - module. Then M contains a submodule isomorphic to R /P for exactly one prime ideal P.
Let F = {{n} : n ∈ Z} and G = {{r} : r ∈ R}. Find the smallest σ-algebra in Z containing F and the smallest σ-algebra in R containing G.
Abstract algebra 2. Graduate level. Note: let R be any ring and M and N R − modules