nilpotent

Thân thiện

A mathematician writes a nilpotent matrix on the chalkboard.

Definition

1. 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).

Học thuật

Adjective

1. equal to zero when raised to a certain power