Please answer the question in the image shown below: **Problem 17:**Prove that the ring \( \mathbb{Z}_m \times \mathbb{Z}_n \) is not isomorphic to \( \mathbb{Z}_{mn} \) if \( m \) and \( n \) are not relatively prime.**Explanation:**In this problem, you are asked to demonstrate that the direct product of the rings \( \mathbb{Z}_m \) and \( \mathbb{Z}_n \) cannot be isomorphic to the ring \( \mathbb{Z}_{mn} \) under the condition that the integers \( m \) and \( n \) have a common divisor greater than 1. The proof requirements hinge on properties of ring isomorphisms and the significance of relatively prime numbers in modular arithmetic. Ring isomorphisms are structure-preserving bijections between two algebraic structures, implying the two structures have identical algebraic properties. Differences in properties like size and unit elements will play key roles in showing the absence of an isomorphism in this scenario.

Name: Please answer the question in the image shown below: **Problem 17:**Pr
Uploaded: 2024-03-05T11:53:49.000Z
Channel: Bartleby
Description: Please answer the question in the image shown below: **Problem 17:**Prove that the ring \( \mathbb{Z}_m \times \mathbb{Z}_n \) is not isomorphic to \( \mathbb{Z}_{mn} \) if \( m \) and \( n \) are not relatively prime.**Explanation:**In this problem,