MAT3143 Fall 2022 Assignment 4
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MAT3143 Fall 2022
Due Friday 2 December, 10pm.
All answers should include complete justiﬁcations (proofs!).
You may use answers from one question (or one part of a question) in your solutions to other questions.
1. Let R be a unital ring, let G be a group, and let M be a ﬁnitely generated module over RG (the group ring). M becomes a module over R (by viewing R as a subring of RG).
(a) Show that if M is free as an RG-module, then it is also free as an R-module.
(b) Is the converse of (a) true? Give a proof or counterexample.
2. Let K be a ﬁeld and let A e Mn (K). Let M be the K[x]-module Kn, where
f(x) . v := f(A)v .
Let I := ann(M).
(a) Assuming A is diagonalizable, show that for λ e K , I C (x _ λ) if and only if λ is an eigenvalue of A.
(b) Show that if f e K[x] is an irreducible polynomial such that f(A) = 0 then deg(f)ln. (Hint: Show that M induces a module over K[x]/(f).
What kind of ring is K[x]/(f)?)
3. Let R be a ring, let N1 , N2 be modules and let M := N1 o N2 . Show that M has a submodule N isomorphic to N1 , and that M/N N2 .
4. Let R be the ring of inﬁnite matrices with entries is Q (with rows and columns indexed by N), such that each row and column have only ﬁnitely many nonzero entries. (We saw this example in Lecture 16.)
(a) Show that the set M of inﬁnite column vectors (with entries indexed by N) is a module over R, using matrix-column multiplication. (b) Find an explicit basis of size 2 for R, as an R-module.
(c) Show that R (as an R-module) has a basis of size n, for any n = 1, 2, . . . .
(d) Does R have an inﬁnite basis (as an R-module)?