2022-2023 Academic Year Math 711, Algebra( Theory of Modules) Assignment I, 18.10.2022 1. Let π΄ be an additive group. Define π: β€ × π΄ βΆ π΄ by ππ (π > 0) (π = 0) π(π, π) = { 0 −|π|π (π < 0). Show that π΄ is a left β€-module under the multiplication above. 2. Let π be a ring. Show that there is a one-to-one correspondence between left π -modules and representations of π via endomorphism rings of abelian groups. 3. Let π be a ring and let π: π βΆ π be a ring homomorphism. Show that any left π-module π is a left π -module. 4. Let π be a left π -module. Prove that the representation π: π βΆEnd(π) is one-to-one if and only if for each π ∈ π with π ≠ 0 there exists π ∈ π with π ≠ 0. 5. Let π be a left π -module, let π be a nonempty subset of π, and let π΄ be a left ideal of π . (a) Show that π΄π₯ = {∑ππ=1 ππ π₯π | ππ ∈ π΄π , π₯π ∈ π, π ∈ β€+ }