Module over a ring

Definition

A module over a ring is a mathematical structure that generalizes the concept of vector spaces by allowing scalars to come from a ring instead of a field. In this context, modules retain many properties of vector spaces but also exhibit unique features due to the properties of rings. Modules play a crucial role in understanding algebraic structures and are particularly important in the study of normal and Cohen-Macaulay varieties, as well as in primary decomposition and associated primes.

5 Must Know Facts For Your Next Test

1. Modules over a ring can be thought of as a way to extend the linear algebra framework, allowing operations with coefficients from rings rather than just fields.
2. The notion of torsion in modules refers to elements that become zero when multiplied by some non-zero element from the ring, which does not occur in vector spaces.
3. The classification of finitely generated modules can be complex and is closely tied to concepts like primary decomposition and associated primes.
4. Normal varieties can often be characterized through their coordinate rings being integrally closed in their field of fractions, revealing connections with module theory.
5. Cohen-Macaulay rings have properties that make their modules behave nicely, such as having consistent depth across all finitely generated modules.

Review Questions

• How do modules over a ring differ from vector spaces, particularly in relation to their structural properties?
  • Modules over a ring differ from vector spaces primarily in that their scalars come from a ring instead of a field. This means that operations within a module can involve zero divisors and non-invertible elements, leading to unique properties such as torsion. Additionally, while every vector space is a module over a field, not every module possesses the same level of simplicity or behavior due to the more complex nature of rings.
• In what ways do primary decomposition and associated primes connect to the study of modules over rings?
  • Primary decomposition involves breaking down ideals into primary components that can be related back to modules. Each component corresponds to associated primes, which describe how modules behave regarding their annihilators. Understanding these connections helps elucidate how finitely generated modules over Noetherian rings can be analyzed in terms of their structure and properties, leading to insights about dimensions and singularities in varieties.
• Evaluate how the properties of Cohen-Macaulay varieties reflect the characteristics of modules over rings in algebraic geometry.
  • Cohen-Macaulay varieties exhibit depth consistency across their coordinate rings, paralleling properties found in modules over Cohen-Macaulay rings. This relationship implies that the depth of finitely generated modules aligns with geometric properties such as smoothness and singularity. By studying these varieties through their associated modules, one can gain deeper insights into how algebraic properties manifest geometrically, which enhances our understanding of both algebraic geometry and commutative algebra.

Related terms

Ring: A set equipped with two binary operations, usually referred to as addition and multiplication, satisfying certain axioms such as associativity and distributivity.

Submodule: A subset of a module that is itself a module under the same operations, maintaining the structure defined by the original module.

Homomorphism: A structure-preserving map between two algebraic structures, such as rings or modules, that respects the operations defined on those structures.