Nilpotent algebra

An associative algebra $A$ over a commutative ring $R$ is defined to be a nilpotent algebra if and only if there exists some positive integer $n$ such that $0=y\_1\backslash \; y\_2\backslash \; \backslash cdots\backslash \; y\_n$ for all $y\_1,\; \backslash \; y\_2,\; \backslash \; \backslash ldots,\backslash \; y\_n$ in the algebra $A$. The smallest such $n$ is called the index of the algebra $A$.{{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}} In the case of a non-associative algebra, the definition is that every different multiplicative association of the $n$ elements is zero.

A power associative algebra in which every element of the algebra is nilpotent is called a nil algebra.[http://www.encyclopediaofmath.org/index.php/Nil_algebra Nil algebra – Encyclopedia of Mathematics]

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.

• Algebraic structure (a much more general term)
• nil-Coxeter algebra
• Lie algebra
• Example of a non-associative algebra

{{reflist}}

• {{Lang Algebra}}

• [http://www.encyclopediaofmath.org/index.php/Nilpotent_algebra Nilpotent algebra – Encyclopedia of Mathematics]

Category:Ring theory

Category:Properties of binary operations