In mathematics, specifically in ring theory, a '''nilpotent algebra over a commutative ring''' is an algebra over a commutative ring, in which for some positive integer ''n'' every product containing at least ''n'' elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon the Lie bracket. (There is no Lie bracket for many algebras over commutative rings; a Lie algebra involves its Lie bracket, whereas, there is no Lie bracket defined in the general case of an algebra over a commutative ring.) Another possible source of confusion in terminology is the ''quantum nilpotent algebra'',<ref>{{cite arXiv|author=Goodearl, K. R.|author2=Yakimov, M. T.|title=Unipotent and Nakayama automorphisms of quantum nilpotent algebras|date=1 Nov 2013|class=math.QA|eprint=1311.0278}}</ref> a concept related to quantum groups and Hopf algebras.
==Formal definition== An associative algebra <math>A</math> over a commutative ring <math>R</math> is defined to be a '''nilpotent algebra''' if and only if there exists some positive integer <math>n</math> such that <math>0=y_1\ y_2\ \cdots\ y_n</math> for all <math>y_1, \ y_2, \ \ldots,\ y_n</math> in the algebra <math>A</math>. The smallest such <math>n</math> is called the '''index''' of the algebra <math>A</math>.<ref>{{cite book|author=Albert, A. Adrian|author-link=A. A. Albert|title=Structure of Algebras|page=22|chapter=Chapt. 2: Ideals and Nilpotent Algebras|orig-year=1939|year=2003|series=Colloquium Publications, Col. 24|publisher=Amer. Math. Soc.|chapter-url=https://books.google.com/books?id=1G0HcOcoJ1cC&pg=PA22|isbn=0-8218-1024-3|issn=0065-9258|postscript=; reprint with corrections of revised 1961 edition}}</ref> In the case of a non-associative algebra, the definition is that every different multiplicative association of the <math>n</math> elements is zero.
==Nil algebra== A power associative algebra in which every element of the algebra is nilpotent is called a ''nil algebra''.<ref>[http://www.encyclopediaofmath.org/index.php/Nil_algebra Nil algebra – Encyclopedia of Mathematics]</ref>
Nilpotent algebras are trivially nil, whereas nil algebras may not be nilpotent, as each element being nilpotent does not force products of distinct elements to vanish.
==See also== * Algebraic structure (a much more general term) * nil-Coxeter algebra * Lie algebra * Example of a non-associative algebra
==References== {{reflist}} *{{Lang Algebra}}
==External links== *[http://www.encyclopediaofmath.org/index.php/Nilpotent_algebra Nilpotent algebra – Encyclopedia of Mathematics]
Category:Algebras Category:Ring theory Category:Properties of binary operations