“School of Mathematics”

Back to Papers Home
Back to Papers of School of Mathematics

Paper   IPM / M / 17340
School of Mathematics
  Title:   Several characterizations of left Kothe rings
1.  Mahmood Behboodi
2.  Shadi Asgari (Joint with S. Khedrizadeh)
  Status:   Published
  Journal: Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.
  Year:  2023
  Pages:   1-17
  Supported by:  IPM
We study classical K�?�¶the�¢??s problem, concerning the structure of non-commutative rings with the property that: �¢??every leftmodule is a direct sum of cyclicmodules".In 1934, K�?�¶the showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring R is called a left K�?�¶the ring if every left R-module is a direct sum of cyclic R-modules. In 1951, Cohen and Kaplansky proved that all commutative K�?�¶the rings are Artinian principal ideal rings. During the years 1961�¢??1965, Kawada solved K�?�¶the�¢??s problem for basic finitedimensional algebras: Kawada�¢??s theorem characterizes completely those finite-dimensional algebras for which any indecomposable module has a square-free socle and a square-free top, and describes the possible indecomposable modules. But, so far, K�?�¶the�¢??s problem is open in the non-commutative setting. In this paper, we classified left K�?�¶the rings into three classes one contained in the other: left K�?�¶the rings, strongly left K�?�¶the rings and very strongly left K�?�¶the rings, and then, we solve K�?�¶the�¢??s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. Finally, we give a new generalization of K�?�¶the�¢??Cohen�¢??Kaplansky theorem.

Download TeX format
back to top
scroll left or right