Annihilator ideal

Definition

An annihilator ideal is a specific type of ideal in a ring that consists of all elements that 'annihilate' a given subset of a module, meaning they send every element of that subset to zero. This concept is closely tied to the structure of cyclic modules, where the annihilator can provide important insights into how elements of the ring interact with elements of the module, influencing properties like simplicity and reducibility.

5 Must Know Facts For Your Next Test

1. The annihilator ideal of a subset \( S \) in a module \( M \) over a ring \( R \) is denoted as \( Ann(S) = \{ r \in R : r \cdot s = 0 \text{ for all } s \in S \} \).
2. For cyclic modules generated by an element \( m \), the annihilator ideal plays a key role in determining the relationship between the generator and other elements in the module.
3. If the annihilator ideal of a cyclic module is non-zero, it indicates that there are non-trivial relationships between the ring elements and the module elements.
4. The concept of annihilator ideals helps in understanding the structure and classification of modules, particularly when analyzing direct sums or decompositions.
5. In the context of finite modules over principal ideal domains (PIDs), the annihilator ideal can provide insights into the torsion properties of modules.

Review Questions

• How does the annihilator ideal relate to the structure of cyclic modules?
  • The annihilator ideal provides critical information about how elements from the ring interact with a generator of a cyclic module. Since cyclic modules can be generated by a single element, understanding which ring elements map this generator to zero helps clarify relationships within the module. If an element belongs to the annihilator ideal, it indicates that multiplying this element with any scalar multiple of the generator results in zero, thereby showing how certain properties may affect the module's overall structure.
• Discuss the implications of having a non-zero annihilator ideal in a cyclic module.
  • When a cyclic module has a non-zero annihilator ideal, it implies there are non-trivial relationships between elements of the ring and those in the module. This means some elements in the ring do not contribute to generating new elements in the module but instead reduce them to zero. Such characteristics can impact factors like simplicity, reducibility, and the overall classification of the module within its broader algebraic context.
• Evaluate how the concept of annihilator ideals aids in classifying modules, especially in terms of torsion properties.
  • The concept of annihilator ideals is crucial for classifying modules, particularly when examining their torsion properties over rings like principal ideal domains (PIDs). By analyzing these ideals, mathematicians can identify whether certain elements are torsion or free, influencing how modules decompose into simpler components. The presence and nature of an annihilator ideal reveal essential structural aspects of modules, guiding mathematicians towards understanding their behavior under various operations and mappings.

Related terms

Cyclic Module: A cyclic module is a module that can be generated by a single element, meaning every element of the module can be expressed as a scalar multiple of this generator.

Ideal:

An ideal is a special subset of a ring that is closed under addition and closed under multiplication by any element in the ring.

Homomorphism:

A homomorphism is a structure-preserving map between two algebraic structures, such as rings or modules, that respects their operations.