Question

Are the two rings \(\mathbb{Z}_{5} \times \mathbb{Z}_{12}\) and \(\mathbb{Z}_{3} \times \mathbb{Z}_{20}\) isomorphic?

Short answer

No, the two rings are not isomorphic.

Step-by-step solution

Understand the Definition of Isomorphism

For two rings to be isomorphic, there exists a bijective ring homomorphism between them that preserves both the addition and multiplication operations.

Assess the Order of Rings

Calculate the number of elements in each ring. The ring \(\mathbb{Z}_5 \times \mathbb{Z}_{12}\) has \(5 \times 12 = 60\) elements. Similarly, \(\mathbb{Z}_3 \times \mathbb{Z}_{20}\) has \(3 \times 20 = 60\) elements. So, both rings have the same number of elements.

Check Structure of Subrings

Consider the structure of elements in each ring. \(\mathbb{Z}_5\) and \(\mathbb{Z}_3\) are fields; \(\mathbb{Z}_{12}\) and \(\mathbb{Z}_{20}\) are not fields since neither is a prime modulus.

Prime Factorization of Moduli

Perform prime factorization: \(12 = 2^2 \times 3\) and \(20 = 2^2 \times 5\). Note that the largest prime factors are different \(3\) for \(12\) and \(5\) for \(20\), which suggests a difference in prime structure.

Analyze Direct Product Structure

The direct product \(\mathbb{Z}_5 \times \mathbb{Z}_{12}\) behaves differently in its element structure than \(\mathbb{Z}_3 \times \mathbb{Z}_{20}\) because of their differing maximal orders \((5 \text{ and } 12) \text{ vs } (3 \text{ and } 20)\). This affects potential isomorphisms.

Conclusion on Isomorphism

Since the rings have different prime factor structures and behavior due to differing moduli, they cannot be isomorphic, as isomorphism requires structural equivalence in both operations and order.

Key concepts

Ring Theory

Ring theory is a branch of abstract algebra that focuses on the study of rings. A ring is a mathematical structure consisting of a set equipped with two operations: addition and multiplication. These operations must satisfy certain properties, such as being associative and distributive. Rings can vary in complexity, from simple integer rings to more complex constructs, like polynomial rings.
A crucial aspect of ring theory is the study of substructures within rings, known as ideals and subrings. These are similar to subgroups in group theory, allowing for deeper insight into the ring's structure.

• In ring theory, an isomorphism between two rings occurs when there's a bijective mapping that preserves the ring operations of addition and multiplication.
• This concept of isomorphism is vital because it shows that two rings are equivalent in structure, even if they seem different superficially.

Understanding these fundamental properties helps in analyzing if two rings can be considered structurally identical through what is known as a ring isomorphism.

Direct Product of Rings

The direct product of rings is an operation that combines two rings to form a new ring. It is denoted as \( R \times S \), where \( R \) and \( S \) are individual rings. This product allows each element of the new ring to be a pair consisting of one element from each of the original rings.
In this context, each operation, addition and multiplication, is performed component-wise. Thus, for elements \((r_1, s_1)\) and \((r_2, s_2)\) in the direct product \( R \times S \):

• The addition is defined as \((r_1 + r_2, s_1 + s_2)\).
• Multiplication is defined as \((r_1 \times r_2, s_1 \times s_2)\).

This direct product maintains the properties of a ring, holding to the associative, distributive, and identity properties for its defined operations. In the analysis of isomorphisms, examining the direct product structure of rings is essential. Whether the rings can map onto each other while preserving structure depends heavily on these component-wise functions and the underlying properties they reveal.

Prime Factorization

Prime factorization involves expressing a number as a product of its prime numbers. It's a critical concept in number theory and plays a significant role in understanding the structure and properties of mathematical objects, such as rings.
When analyzing whether two rings are isomorphic, examining the prime factorization of their orders can reveal differences in structure. This is because isomorphism often requires structural similarity, and prime factorization provides insight into the composition of elements in a ring.

• For instance, in our problem, the moduli \(12 = 2^2 \times 3\) and \(20 = 2^2 \times 5\) have different prime structures.
• The largest prime factors, \(3\) and \(5\), suggest differences in their respective ring structures.

This structural difference indicates that a bijective structure-preserving mapping, necessary for isomorphism, is unlikely to exist because the foundation of each ring (its prime composition) is inherently different. Hence, prime factorization is a powerful tool in distinguishing whether two rings can be isomorphic or not.