Ring of Integers

Definition

The ring of integers, denoted as $$ ext{Z}$$, is the set of all whole numbers that can be expressed as positive or negative whole numbers, including zero. This structure is crucial in commutative algebra, particularly as it serves as a fundamental example of a Noetherian ring, which is essential for understanding properties of rings and modules, as the ring of integers has unique factorization and satisfies the ascending chain condition on ideals.

5 Must Know Facts For Your Next Test

1. The ring of integers is an example of a Noetherian ring because every ideal in $$ ext{Z}$$ is finitely generated.
2. The integers form a unique factorization domain, which means every non-zero integer can be uniquely factored into prime numbers.
3. In the context of modules over the ring of integers, every finitely generated module is also finitely presented.
4. The only prime ideals in the ring of integers are generated by prime numbers and the zero ideal.
5. The ring of integers has a rich structure that includes both algebraic and topological properties, making it an essential object of study in various areas of mathematics.

Review Questions

• How does the ring of integers exemplify the properties of Noetherian rings?
  • The ring of integers demonstrates the properties of Noetherian rings through its ideal structure. Every ideal in $$ ext{Z}$$ can be generated by a single integer, which indicates that there are no infinite ascending chains of ideals; thus, it satisfies the ascending chain condition. This property makes the ring of integers an important example when studying Noetherian rings and helps to establish foundational concepts in commutative algebra.
• Discuss the implications of unique factorization in the ring of integers and its significance in algebra.
  • Unique factorization in the ring of integers means that every integer greater than one can be expressed uniquely as a product of prime numbers. This property is significant because it establishes a strong link between number theory and algebra. It allows for various algebraic techniques to be applied to solve problems related to divisibility and modular arithmetic, making it a crucial aspect in understanding both integer properties and broader algebraic structures.
• Evaluate how the structure of the ring of integers influences its modules, specifically focusing on finitely generated modules.
  • The structure of the ring of integers significantly influences its modules, particularly in that any finitely generated module over $$ ext{Z}$$ can be expressed as a direct sum of cyclic modules. This leads to a classification theorem for finitely generated abelian groups, showing that they can be decomposed into torsion-free parts and finite cyclic components. Thus, understanding the properties of the ring of integers provides deep insights into the behavior and characteristics of its associated modules.

Related terms

Noetherian Ring:

A ring that satisfies the ascending chain condition on ideals, meaning that every increasing sequence of ideals eventually stabilizes.

Unique Factorization Domain (UFD): A type of integral domain in which every non-zero element can be factored uniquely into irreducible elements, up to order and units.

Ideal:

A special subset of a ring that absorbs multiplication by elements from the ring and allows for the construction of quotient rings.