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Left Co-Kothe and Generalized Co-Kothe Rings and Solving an Old Problem of Kothe on Non-Commutative Rings | | Shadi Asgari
; Mahmood Behboodi
; Somayeh Khedrizadeh
; | | Date: |
13 Jun 2022 | | Abstract: | We solve the classical Kothes problem, concerning the structure of
non-commutative rings with the property that: every left module is a direct sum
of cyclic modules. A ring $R$ is left (resp., right) Kothe if every left
(resp., right) R-module is a direct sum of cyclic $R$-modules. Kothe [Math. Z.
39 (1934), 31-44] showed that all Artinian principal ideal rings are left Kothe
rings. Cohen and Kaplansky [Math.Z 54 (1951), 97-101] proved that all
commutative Kothe rings are Artinian principal ideal rings. Faith [Math. Ann.
164 (1966), 207-212] characterized all commutative rings whose proper factor
rings are Kothe rings. However, Nakayama [Proc. Imp. Acad. Japan 16 (1940),
285-289] gave an example of a left Kothe ring which is not a principal ideal
ring. Kawada [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 7 (1962), 154-230; Sect.
A 8 (1965),165-250] completely solved Kothes problem for basic finite
dimensional algebras. So far the Kothe’s problem was still open in the
non-commutative setting. In this paper, among other related results, we solve
Kothes problem for any ring. We also determine the structure of left co-Kothe
rings (rings whose left modules are direct sums of co-cyclics modules).
Finally, as an application, we present several characterizations of left Kawada
rings, which generalizes a well-known result of Ringel on finite dimensional
Kawada algebras (a ring R is called left Kawada if any ring Morita equivalent
to R is a left Kothe ring). | | Source: | arXiv, 2206.06453 | | Services: | Forum | Review | PDF | Favorites | |
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