nilpotent
Definition
- Adjective:
- Equal to zero when raised to a certain power: In mathematics, particularly in ring theory and linear algebra, an element (like a matrix or an operator) is described as "nilpotent" if there exists some positive integer power to which it can be raised that results in zero.
Usage Examples
- Adjective:
- In the ring of 2x2 matrices, the matrix with a 1 in the top-right corner and zeros elsewhere is nilpotent because its square is the zero matrix.
- A nilpotent element in a ring must have the property that some finite power of it equals the additive identity, zero.
- The linear operator was proven to be nilpotent, meaning repeated application eventually maps any vector to the zero vector.
Advanced Usage
- Nilpotent Group: In group theory, a group where the lower central series eventually reaches the trivial subgroup.
- The study of nilpotent groups is important in understanding the structure of finite groups.
- Nilpotent Ideal: An ideal in a ring where every element is nilpotent.
- The radical of a ring is often a nilpotent ideal.
Variants and Related Words
- Nilpotency (n): The property or condition of being nilpotent.
- The nilpotency index of an element is the smallest power needed to get zero.
- Nilpotently (adv): In a nilpotent manner.
- The operator acts nilpotently on the vector space.
Synonyms
- Annihilating (in specific contexts, meaning producing zero).
- Zero-potent (rare, technical synonym).
Related Concepts (Not Phrasal Verbs)
- Idempotent: An element equal to its own square (contrasting concept).
- Zero divisor: An element that can multiply with a non-zero element to produce zero (a related but distinct property).
Adjective
-
equal to zero when raised to a certain power
