13.12.8 Problem. Let R be a ring and, let M be an R-module. Let m be a nonnegative integer, and suppose that M1,..., Mm are R-submodules of M, and that M is the internal direct sum of M1,..., Mm. Let n be a nonnegative integer with n < m, and for each i E {1,...,n}, let N; be an R-submodule of M. Let N = N1 ++ Nn. ... (i) Prove that N is the internal direct sum of N1,...,...
Q: Let L be a finite-dimensional Lie algebra over C with universal enveloping algebra U(L), and let V and W be L-modules. (1) Define what is meant by an L-module homomorphism o: V the modules V and W to be isomorphic W and explain what it means for (ii) Explain what is meant by a submodule S of V and describe the factor module V/S. V W be an L-module homomorphism Let (iii) Show that ker(ø) is a submodule of...
9. Prove that (a - b) x (a+ b) = 2(a x b) X 9. Prove that (a - b) x (a+ b) = 2(a x b) X
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cyclic subgroups of (G, ). (c) Explain why (G, ) is not isomorphic to the symmetric group S3. (d) State an isomorphism between (G, ) and (Z6, +).
Q 3 a) Let n > 2 be an integer. Prove that the set {z ET:z” = 1} is a subgroup of (T, *). Show that it is isomorphic to (Zn, + mod n). b) Show that Z2 x Z2 is not isomorphic to Z4. c) Show that Z2 x Z3 is isomorphic to 26.
1. (2 pts) Prove that C[t] is not isomorphic to Clr.y]/(y2-13) as a C- algebra. (It follows that the affine line A1 is not isomorphic to the variety X CA2 defined by the equation y 1. (2 pts) Prove that C[t] is not isomorphic to Clr.y]/(y2-13) as a C- algebra. (It follows that the affine line A1 is not isomorphic to the variety X CA2 defined by the equation y
Question 9. Prove that if M' is a maximal matching and M is a maximum matching in a graph G, then AMM /2.
Do A and used C as question say A. (This problem gives an explanation for the isomorphism R 1m(A) R"/1m(A'), where A, Q-IAP, with Q and P invertible.) Let R be a ring and let M, N, U, V be R-modules such that there existR module homomorphisms α : M N, β : u--w, γ: M-+ U and δ: N V such that the following diagram is commutative: (recall that commutativity of the diagram means that δ ο α γ)...
9. a) Prove that ifA is invertible and AB-AC then B = C.
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...