simple algebra

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simple algebra

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simple algebra (plural simple algebras)

1. (algebra) An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0).
   • 1945 , E. T. Bell, 2nd Edition, 1992, Dover, The Development of Mathematics, page 252,

     A semi-simple algebra over F is either simple or the direct sum of simple algebras over F.

   • 1989 , Nathan Jacobson, Basic Algebra II, 2nd Edition, 2009, Dover, page 215,

     If ${\displaystyle A}$ is a finite dimensional simple algebra over ${\displaystyle F}$, then ${\displaystyle A\cong M_{n}(\Delta )}$ where ${\displaystyle \Delta }$ is a finite dimensional division algebra over ${\displaystyle \Delta }$.

   • 2005, Igor R. Shafarevich, translated by Miles Reid, Basic Notions of Algebra, Springer, page 198:

     One proves that if ${\displaystyle {\mathcal {L}}}$ is a simple algebra over ${\displaystyle \mathbb {R} }$, then ${\displaystyle L_{C}}$ is either a simple algebra over ${\displaystyle \mathbb {C} }$, or a direct sum of two isomorphic simple algebras. Thus the problem of studying simple algebras over ${\displaystyle \mathbb {R} }$ reduces to a similar problem over ${\displaystyle \mathbb {C} }$.

• central simple algebra

• semisimple algebra

• Central simple algebra on Wikipedia.Wikipedia
• Semisimple algebra on Wikipedia.Wikipedia