# Can Direct Sum Isomorphism Imply Module Equality in PID?

• bruno321
In summary, the conversation is discussing the "little fact" that if M and N are finitely generated modules over a PID and M+M=N+N, then M=N. The speaker believes it can be proven using the structure theorem, but it turns out to not be true in general. The structure theorem is mentioned as a potential approach to proving the fact.
bruno321
Hi. I'm trying to prove this "little fact": let M, N be finitely generated modules over a PID. Then if M+M=N+N (where = means isomorphism and + means direct sum) then M=N.

I'm sure it can be done with the structure theorem (it is obvious from the hypotheses); it looks like it should be trivially proven, but alas, I don't think it can be.

What do you think?

Cheers,

EDIT: this isn't true the way I originally thought it was...thinking harder!

Well it's certainly not true without the finitely generated over a PID hypothesis (just think about free abelian groups of infinite rank), so I'm guessing it's some particular property of finitely generated modules over PIDs. The structure theorem immediately comes to mind.

M + m = n + n.
2m = 2n
m = n

## 1. What is the equation "M+M = N+N then M=N" referring to?

The equation is referring to the concept of modular arithmetic, where the values on each side of the equation are congruent (equivalent) in a specific modular system.

## 2. Can you give an example of how this equation works in modular arithmetic?

Sure, for example, in the modular system of clock arithmetic where the modulus is 12 (representing the 12 hours on a clock), if we have 3 o'clock + 3 o'clock = 6 o'clock + 6 o'clock, then we can say that 3 is congruent to 6 (mod 12), or in other words, 3 is equivalent to 6 in this modular system.

## 3. How is this equation relevant in real-world applications?

The concept of modular arithmetic has many practical applications, such as in computer programming, cryptography, and scheduling systems. It allows for efficient handling and manipulation of large numbers and can also be used for data encryption.

## 4. Is this equation always true in all modular systems?

No, this equation is only true in specific modular systems where the modulus is equal on both sides. For example, in the modular system of clock arithmetic with a modulus of 12, the equation holds true, but in a different modular system with a different modulus, it may not hold true.

## 5. How is this equation different from regular arithmetic?

In regular arithmetic, the values on each side of the equation must be equal in order for the equation to be true. However, in modular arithmetic, the values must be congruent in a specific modular system for the equation to hold true.

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