Nilpotent algebra

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, a concept related to quantum groups and Hopf algebras. 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, a concept related to quantum groups and Hopf algebras. An associative algebra A {displaystyle A} over a commutative ring R {displaystyle R} is defined to be a nilpotent algebra if and only if there exists some positive integer n {displaystyle n} such that 0 = y 1   y 2   ⋯   y n {displaystyle 0=y_{1} y_{2} cdots y_{n}} for all y 1 ,   y 2 ,   … ,   y n {displaystyle y_{1}, y_{2}, ldots , y_{n}} in the algebra A {displaystyle A} . The smallest such n {displaystyle n} is called the index of the algebra A {displaystyle A} . In the case of a non-associative algebra, the definition is that every different multiplicative association of the n {displaystyle n} elements is zero. An algebra in which every element of the algebra is nilpotent is called a nil algebra.