associated prime
The concept of 'associated prime' is fundamental in algebra, particularly in module theory and commutative algebra, and has implications in algebraic geometry.
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Definition
C2Module Theory
(technical, academic)A prime ideal in a ring that serves as the annihilator of a prime submodule of a given module.
Example
- In module theory, identifying the associated primes can provide insights into the structure of the module.
C2Commutative Algebra
(technical, academic)A prime ideal in a commutative ring that is the annihilator of some element in a module over that ring.
Example
- The associated primes of a module over a commutative ring help in understanding the module's decomposition.
C2Algebraic Geometry
(technical, academic)A prime ideal in a ring that corresponds to a point in the spectrum of the ring, linked to a module or sheaf.
Example
- In algebraic geometry, associated primes can indicate the geometric properties of a module.
Similar
Terms that have similar or relatively close meanings to "associated prime":