Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-28T21:45:37.840Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALIZED HIGHER DERIVATIONS

Part of: Rings and algebras arising under various constructions Differential algebra

Published online by Cambridge University Press:  06 January 2012

E. P. COJUHARI  and
B. J. GARDNER
E. P. COJUHARI*
Affiliation:
Department of Mathematics, Technical University of Moldova, Ştefan cel Mare av. 168, Chişinău, MD-2004, Moldova (email: [email protected])
B. J. GARDNER
Affiliation:
Discipline of Mathematics, University of Tasmania, Private Bag 37, Hobart, Tas. 7001, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A type of generalized higher derivation consisting of a collection of self-mappings of a ring associated with a monoid, and here called a D-structure, is studied. Such structures were previously used to define various kinds of ‘skew’ or ‘twisted’ monoid rings. We show how certain gradings by monoids define D-structures. The monoid ring defined by such a structure corresponding to a group-grading is the variant of the group ring introduced by Năstăsescu, while in the case of a cyclic group of order two, the form of the D-structure itself yields some gradability criteria of Bakhturin and Parmenter. A partial description is obtained of the D-structures associated with infinite cyclic monoids.

Keywords

derivationhigher derivationgraded ringmonoid algebra

MSC classification

Secondary: 13N15: Derivations 16S36: Ordinary and skew polynomial rings and semigroup rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 266 - 281
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Abu-Saymeh, S. and Ikeda, M., ‘On the higher derivations of communitive [sic] rings’, Math. J. Okayama Univ. 29 (1987), 83–90.Google Scholar
[2]Bahturin, Yu. A and Parmenter, M. M., ‘Group gradings on integral group rings’, in: Groups, Rings and Group Rings, Lecture Notes in Pure and Applied Mathematics, 248 (eds. Giambruno, A., Polcino Milies, C. and Sehgal, S. K.) (Chapman & Hall/CRC, Boca Raton, FL, 2006), pp. 25–32.CrossRefGoogle Scholar
[3]Cojuhari, E. P., ‘Monoid algebras over noncommutative rings’, Int. Electron. J. Algebra 2 (2007), 28–53.Google Scholar
[4]Hasse, H. and Schmidt, F. K., ‘Noch eine Begründung der Theorie der höheren Differentialquotienten einem algebraischen Funktionenkörper einer Unbestimmten’, J. reine angew. Math. 177 (1937), 215–237.Google Scholar
[5]Heerema, N., ‘Derivations and embeddings of a field in its power series ring’, Proc. Amer. Math. Soc. 11 (1960), 188–194.CrossRefGoogle Scholar
[6]Miller, J. B., ‘Homomorphisms, higher derivations, and derivations’, Acta Sci. Math. (Szeged) 28 (1967), 221–231.Google Scholar
[7]Năstăsescu, C., ‘Group rings of graded rings. Applications’, J. Pure Appl. Algebra 33 (1984), 313–335.CrossRefGoogle Scholar
[8]Parmenter, M. M., personal communication.Google Scholar
[9]Smits, T. H. M., ‘Nilpotent S-derivations’, Indag. Math. 30 (1968), 72–86.CrossRefGoogle Scholar