simple algebra

English

Noun

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

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

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

Hyponyms

• central simple algebra

Translations

algebra that contains no nontrivial proper ideals and whose multiplication operation is not zero

• Italian: algebra semplice f

See also

• semisimple algebra

Further reading

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