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abelian group algebraically closed field artinian Chosun University coefficient ring commutative ring completes the proof converse statement Corollary dim N(KG divides the order element of G field of ch field whose characteristic finite normal subgroups finite subgroup finitely generated abelian finitely generated subgroup following theorem forms a basis G is locally group G group of order group ring KG H of G ideal of KG implies infinite cyclic group integers irreducible isomorphic J(KG J(KH Jacobson radical K-basis K-module Ker.f KG is semi-prime KG is semi-simple Lemma Let G Let H locally finite groups m-system maximal ideal modular representation theory nilpotent elements nilpotent ideal normal p-Sylow subgroup order pn p-element prime ideal prime radical principal indecomposable module Proposition 1-1 prove the following radical of KG set of orders subdirect product subgroup H subgroup of G subgroup of order submodule Supp(x Suppose Theorem 1-3 totally ordered groups x£KG