I don't have any help for you but I have to comment that your post gave me a chuckle because my son and I were just recently discussing a linguistic phenomenon that I noticed some time ago. I make no representation that I know what YOU mean, but I know what I mean, and what most people mean with the following construct:Mr Davis 97 said:
Two rings are considered isomorphic if there exists a bijective ring homomorphism between them. This means that the two rings have the same structure and operations, but the elements may be named differently.
If two rings are isomorphic, they must have the same number of elements, the same identity element, and the same operations (addition and multiplication) defined on their elements. They also have the same number of zero divisors and nilpotent elements.
No, two rings with different structures cannot be isomorphic. Isomorphism depends on the structure and operations of the rings, so if these are not the same, the rings cannot be isomorphic.
One common technique is to look at the properties of the rings, such as the number of elements, the presence of zero divisors, and the existence of certain elements (like nilpotent elements). If these properties are different, the rings are not isomorphic. Another technique is to find a specific property or feature of one ring that is not present in the other, which would also prove that they are not isomorphic.
There are no shortcuts or tricks to proving that two rings are not isomorphic. It requires a thorough understanding of the properties and structure of both rings, as well as a careful analysis of their differences. It is also important to be familiar with the definition of isomorphism and how it applies to rings.