In
mathematics, specifically in
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, a nilpotent algebra over a commutative ring is an
algebra over a commutative ring
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...
, 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
In mathematics, a Lie algebra \mathfrak is nilpotent if its lower central series terminates in the zero subalgebra. The ''lower central series'' is the sequence of subalgebras
: \mathfrak \geq mathfrak,\mathfrak\geq mathfrak, has a different definition, which depends upon the Lie bracket">mathfrak,\mathfrak ...
has a different definition, which depends upon the Lie bracket. (There is no Lie bracket for many algebras over commutative rings; a
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'', a concept related to
s.
is defined to be a nilpotent algebra if and only if there exists some positive integer
. The smallest such
. In the case of a
elements is zero.
is called a ''nil algebra''.