2002 Fiscal Year Final Research Report Summary
Fundamental research on modules of differentials of noncommutative algebras and its application
| Project/Area Number |
13640040
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Okayama Prefectural University |
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Principal Investigator |
KOMATSU Hiroaki Okayama Prefectural University, Faculty of Computer Science and System Engineering, Associate Professor, 情報工学部, 助教授 (10178361)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAJIMA Atsushi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (30032824)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Module of differentials / Differential operator / Generalized derivation / Ring extension / Matrix ring / Prime ring |
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| Research Abstract |
We constructed the module of high order differentials which determines the structure of all high order derivations, and we studied the theory related them.
We generalized the concept of the high order one-sided derivations and the concept of the module of high order differentials to noncommutative ring extensions. By making use of the two-sided module structure of the module of high order differentials, we detected that the module of high order differentials represents high order one-sided derivations and certain derivations which are called high order central derivations. Applying this fact we got the following noteworthy results : (1) The module of high order differentials can be decomposed by any idempotent element. Even if we consider the algebra case, modules of high order differentials of ring extensions can appear as components of decomposition by idempotent elements. Hence, the concept of the module of high order differentials of ring extensions is significant. (2) Some fundamental exact sequences of modules of high order differentials were given which are unknown in commutative algebras. As applications, we got some results on separable extensions and purely inseparable algebras.
Furthermore, we gave a method to make high order derivations for noncommutative ring extensions. High order one-sided derivations are the special cases of our high order derivations. We constructed their module of high order differentials and gave some fundamental exact sequences of module of high order derivations.
Our differential operators of order (1, 1) coincides with generalized derivations. We determined rings having a generalized derivation whose nonzero value is always invertible. We also studied generalized derivations of rings without identity element. We determined all generalized derivations of full matrix rings, and we got some results on generalized derivations of prime rings.
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