Solving I^m + J^n = R with Positive Integers m, n

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In summary, to show that there exist positive numbers m,n such that I^m + J^n = R, the trick is to show that I^m + J^n contains 1 and then find a^m in I^m and b^n in J^n such that a^m + b^n = 1. This can be achieved by realizing that a typical element in I^n is not of the form a^n for a in I, but a sum of n'th powers of elements in I.
  • #1
Pietjuh
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Suppose we have a commutative ring R and ideals I and J of R such that I + J = R. I have to show that there exist positive numbers m,n such that I^m + J^n = R.

I think the trick is just to show that I^m + J^m contains 1. Because I + J = R, I+J contains 1 so there exist i in I and j in J such that i + j = 1. Now I have to find a a^m in I^m and a b^n in J^n such that a^m + b^n = 1.
I tried a lot of things but none of them seemed to work :(

Can anyone give me a hint how to find these a^m and b^n ?

Thanks in advance
 
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  • #2
Why isn't n=m=1 sufficient for your purpose?

It might help you to realize that a typical element in I^n is not of the form a^n for a in I, but a sum of n'th powers of elements in I
 
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1. What is the meaning of "solving I^m + J^n = R with positive integers m, n"?

This means finding values for the variables m and n that satisfy the equation I^m + J^n = R, where I, J, and R are constants. The values of m and n must be positive integers, meaning whole numbers greater than zero.

2. What are some common techniques for solving equations with positive integer solutions?

Some common techniques include trial and error, substitution, and using algebraic properties such as factoring and expanding. Another useful approach is to break the problem down into smaller parts and solve each part individually.

3. Are there any restrictions on the values of I, J, and R in this equation?

No, there are no restrictions on the values of I, J, and R as long as they are constants. However, it is important to note that the equation may not have a solution if the values are too large or if they do not follow a pattern.

4. How can I check if my solution for m and n is correct?

You can check your solution by plugging in the values for m and n into the original equation and seeing if it equals the given value of R. Another way to check is by using a calculator to evaluate the left side of the equation and comparing it to the value of R.

5. Is there a specific method or formula for solving these types of equations?

There is no one specific method or formula for solving equations with positive integer solutions. It often requires a combination of different techniques and strategies, and the approach may vary depending on the specific equation. It is important to be creative and flexible when trying to find solutions for these types of equations.

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