A generalization of Mathieu subspaces to modules of associative algebras

Wenhua Zhao

Open Mathematics (2010)

  • Volume: 8, Issue: 6, page 1132-1155
  • ISSN: 2391-5455

Abstract

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We first propose a generalization of the notion of Mathieu subspaces of associative algebras 𝒜 , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to 𝒜 -modules ℳ . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of 𝒜 -modules ℳ , where R is the base ring of 𝒜 . We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.

How to cite

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Wenhua Zhao. "A generalization of Mathieu subspaces to modules of associative algebras." Open Mathematics 8.6 (2010): 1132-1155. <http://eudml.org/doc/269229>.

@article{WenhuaZhao2010,
abstract = {We first propose a generalization of the notion of Mathieu subspaces of associative algebras \[ \mathcal \{A\} \] , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to \[ \mathcal \{A\} \] -modules \[ \mathcal \{M\} \] . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of \[ \mathcal \{A\} \] -modules \[ \mathcal \{M\} \] , where R is the base ring of \[ \mathcal \{A\} \] . We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.},
author = {Wenhua Zhao},
journal = {Open Mathematics},
keywords = {Mathieu subspaces of associative algebras; Mathieu subspaces of modules of associative algebras; (quasi-)stable elements; (quasi-)stable algebras; (quasi-)stable modules; Mathieu subspaces of algebras; Mathieu subspaces of modules; strongly simple algebras; quasi-stable algebras; radicals; idempotents; Jacobian conjecture; matrix algebras; quasi-stable elements; quasi-stable modules},
language = {eng},
number = {6},
pages = {1132-1155},
title = {A generalization of Mathieu subspaces to modules of associative algebras},
url = {http://eudml.org/doc/269229},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Wenhua Zhao
TI - A generalization of Mathieu subspaces to modules of associative algebras
JO - Open Mathematics
PY - 2010
VL - 8
IS - 6
SP - 1132
EP - 1155
AB - We first propose a generalization of the notion of Mathieu subspaces of associative algebras \[ \mathcal {A} \] , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to \[ \mathcal {A} \] -modules \[ \mathcal {M} \] . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of \[ \mathcal {A} \] -modules \[ \mathcal {M} \] , where R is the base ring of \[ \mathcal {A} \] . We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.
LA - eng
KW - Mathieu subspaces of associative algebras; Mathieu subspaces of modules of associative algebras; (quasi-)stable elements; (quasi-)stable algebras; (quasi-)stable modules; Mathieu subspaces of algebras; Mathieu subspaces of modules; strongly simple algebras; quasi-stable algebras; radicals; idempotents; Jacobian conjecture; matrix algebras; quasi-stable elements; quasi-stable modules
UR - http://eudml.org/doc/269229
ER -

References

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  22. [22] Zhao W., Willems R., Analogue of the Duistermaat-van der Kallen theorem for group algebras (in preparation) Zbl1255.16022

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