mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, specifically in ring theory, 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 map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...

, 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...

. (There is no Lie bracket for many algebras over commutative rings; a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

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 algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...

Formal definition

associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

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\  y_2\ \cdots\ y_n

for all

y_1, \  y_2, \ \ldots,\ y_n

in the algebra

A

. The smallest such

n

is called the index of the algebra

A

. In the case of a

non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...

, the definition is that every different multiplicative association of the

n

elements is zero.

Nil algebra

power associative In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra (or more generally a magma) is said to be power-associative if the subalgebra g ...

algebra in which every element of the algebra is

nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

is called a ''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.

References

*{{Lang Algebra

External links

Nilpotent algebra – Encyclopedia of Mathematics
Ring theory Properties of binary operations

Formal definition

Nil algebra

See also

References

External links