simple algebra

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 ${\\displaystyle A}$ 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 ${\\displaystyle \\mathbb {C} }$.

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