simple algebra

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 [McGraw-Hill], 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 [W. H Freeman & Company], 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 ${\displaystyle \Delta }$ .^[sic]

   • 2005, Miles Reid (translator), Igor R. Shafarevich, 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 ${\displaystyle \mathbb {C} }$ .

• semisimple algebra

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