(algebra) A nilpotent element.
Used in a Sentence
nilpotent
Definitions, parts of speech, synonyms, and sentence examples for nilpotent.
Editorial note
Dual numbers use nilpotent infinitesimals (which are not invertible), while non-standard (Robinson) infinitesimals are not nilpotent and are invertible.
Examples12
Definitions4
Parts of speech2
Quick take
(algebra) A nilpotent element.
Meaning at a glance
The clearest senses and uses of nilpotent gathered in one view.
(mathematics, algebra, ring theory, of an element x of a ring) Such that, for some positive integer n, xⁿ = 0.
(ring theory, of an ideal I) Such that there exists a natural number k with Iᵏ = 0.
Definitions
Core meanings and parts of speech for nilpotent.
noun
(algebra) A nilpotent element.
adjective
(mathematics, algebra, ring theory, of an element x of a ring) Such that, for some positive integer n, xⁿ = 0.
adjective
(ring theory, of an ideal I) Such that there exists a natural number k with Iᵏ = 0.
adjective
(group theory, of a group) Admitting a central series of finite length.
Example sentences
Dual numbers use nilpotent infinitesimals (which are not invertible), while non-standard (Robinson) infinitesimals are not nilpotent and are invertible.
For example, projection on x axis + 90 ° rotation of a vector is nilpotent.
A ring is of bounded index if there is some N such that aᴺ is zero for every nilpotent a in A.
Then there is a result by Gromov [2] which says that if the growth is polynomial then the group is virtually nilpotent.
The statement is true for a certain class of matrices called nilpotent matrices.
Squaring a upper triangular matrix with 0 on the diagonal is nilpotent.
The dual numbers are not ordered and contain non-invertible nilpotent elements such as h which squares to 0.
Using the dual numbers approach, there is an infinitesimal (nilpotent) region around each point which is linear.
In contrast, all that is necessary here is extending R to an algebra with a nilpotent element (e^2 = 0).
Both have nilpotent elements, but with dual numbers the background logic is classical, whereas it isn't with smooth infinitesimal analysis.
My favorite are nilpotent infinitesimals which give rise to synthetic differential geometry[3] and dual numbers[4].
The equivalent to i = sqrt(-1) might be something like the nilpotent infinitesimal where epsilon^2 = 0 but epsilon =/= 0.
Frequently asked questions
Short answers drawn from the clearest meanings and examples for this word.
How do you use nilpotent in a sentence?
Dual numbers use nilpotent infinitesimals (which are not invertible), while non-standard (Robinson) infinitesimals are not nilpotent and are invertible.
What does nilpotent mean?
(algebra) A nilpotent element.
What part of speech is nilpotent?
nilpotent is commonly used as noun, adjective.