The Study of Kothe’s Conjecture for Some Rings

Abstract

One of the unsolved problems in mathematics especially for the theory of ring and ideal is Kothe’s conjecture. It stated whether if a ring R has no nil ideal except {0} then R has no one-sided nil ideal except {0}. This question is simple, but very complicated to be solved. Mathematicians developed some equivalent statements to Kothe’s conjecture to simplify this conjecture. Although this conjecture has proven for some rings, but until now it still open for general ring, especially for non-commutative ring. The purpose of this paper is to study about Kothe’s conjecture for some rings. Based on literature study and observation, we conclude that Kothe’s conjecture is true for commutative ring. In additional results, we state the counterexample of the invers of Kothe’s conjecture and study more deeply in some non-commutative rings, those are  and matrix ring. The result is positive solution for some spesific case of non-commutative rings.