2022-2023 Academic Year
Math 711, Algebra( Theory of Modules)
Assignment I, 18.10.2022
1. Let 𝐴 be an additive group. Define 𝜇: ℤ × 𝐴 ⟶ 𝐴 by
𝑛𝑎 (𝑛 > 0)
(𝑛 = 0)
𝜇(𝑛, 𝑎) = { 0
−|𝑛|𝑎 (𝑛 < 0).
Show that 𝐴 is a left ℤ-module under the multiplication above.
2. Let 𝑅 be a ring. Show that there is a one-to-one correspondence between left 𝑅-modules
and representations of 𝑅 via endomorphism rings of abelian groups.
3. Let 𝑆 be a ring and let 𝜙: 𝑅 ⟶ 𝑆 be a ring homomorphism. Show that any left 𝑆-module 𝑀
is a left 𝑅-module.
4. Let 𝑀 be a left 𝑅-module. Prove that the representation 𝜌: 𝑅 ⟶End(𝑀) is one-to-one if
and only if for each 𝑟 ∈ 𝑅 with 𝑟 ≠ 0 there exists 𝑚 ∈ 𝑀 with 𝑚 ≠ 0.
5. Let 𝑀 be a left 𝑅-module, let 𝑋 be a nonempty subset of 𝑀, and let 𝐴 be a left ideal of 𝑅.
(a) Show that 𝐴𝑥 = {∑𝑛𝑖=1 𝑎𝑖 𝑥𝑖 | 𝑎𝑖 ∈ 𝐴𝑖 , 𝑥𝑖 ∈ 𝑋, 𝑘 ∈ ℤ+ }