simple algebra

plural simple algebras

1. algebraAn 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.

   2 more examples

   1989 [W. H Freeman & Company], Nathan Jacobson, Basic Algebra II, 2nd Edition, 2009, Dover, page 215, If A is a finite dimensional simple algebra over F, then A≅M_n(Δ) where Δ is a finite dimensional division algebra over Δ.

   One proves that if #92;mathcal#123;L#125; is a simple algebra over #92;mathbb#123;R#125;, then L#95;C is either a simple algebra over #92;mathbb#123;C#125;, or a direct sum of two isomorphic simple algebras. Thus the problem of studying simple algebras over #92;mathbb#123;R#125; reduces to a similar problem over #92;mathbb#123;C#125;.

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