Completely Prime Modules Over Path Algebras for Dynkin Quiver of Type A _n and D _n

Abstract

Let 𝐾 be a field and 𝑄 be a quiver (directed graph), and let 𝐴 = 𝐾𝑄 be the path algebra that corresponds to 𝑄 with coefficients in 𝐾. An 𝐴-module 𝑀 is a completely prime (𝑐-prime) module in the sense that 𝑟𝑚 = 0 for one 𝑚 ∈ 𝑀 and 𝑟 ∈ 𝐴 implies that either 𝑟 annihilates all 𝑀 or 𝑚 = 0. In this paper, we prove that for Dynkin quiver type 𝐴_𝑛 and 𝐷_𝑛 , an 𝐴-module 𝑀 is 𝑐-prime if and only if it is simple.