Idempotents of the ring Z/100Z

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Please read the PF rules before posting any question and use the appropriate section. In summary, to find the idempotents of the ring Z/100Z, one must find elements that satisfy the equation a² = a + k, where k is an integer and a is an element in the range of 0 to 99. More information can be found in the Homework Help section of Physics Forum.
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chivhone
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How do you proove what the idempotents of the ring Z/100Z are?

I know by trial and error that they are the elements 0,1,25,76 but have no proof as to why this is.
i know i have to find the integers 'a' such that 100 divides a(a-1)=a^2 -a but can do the maths to get a proper reasoning

any help?
thanks
 
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Chose an element [a] of Z/100Z whose representant a is such that [itex]0\leq a \leq 99[/itex] (for simplicity). Then by definition, [a] will be idempotent if [a]²=[a]. That is to say, if a²+100Z = a+100Z. But this will happen if and only if there exists a k such that a² = a + k. Can you go from there?

Between, this question belongs to the Homework help section of Physics Forum (PF): https://www.physicsforums.com/forumdisplay.php?f=152
 

1. What are idempotents in the ring Z/100Z?

Idempotents in the ring Z/100Z are elements that satisfy the property a^2 = a, where a is an element of the ring. In simpler terms, an idempotent is an element that, when multiplied by itself, gives the same element.

2. How many idempotents are there in the ring Z/100Z?

There are 10 idempotents in the ring Z/100Z. This can be seen by considering all the possible elements a in the ring and checking which ones satisfy the property a^2 = a.

3. Can there be more than one idempotent in the ring Z/100Z with the same value?

No, there cannot be more than one idempotent in the ring Z/100Z with the same value. This is because the ring Z/100Z is a finite ring, and each element has a unique inverse. Therefore, if two elements have the same value, they would be each other's inverse, but this contradicts the definition of an idempotent.

4. What is the significance of idempotents in the ring Z/100Z?

Idempotents in the ring Z/100Z have various mathematical applications, such as in coding theory and algebraic geometry. They also have a connection to the Chinese Remainder Theorem, which states that the ring Z/nZ can be decomposed into smaller rings, where n is a product of prime numbers. Idempotents play a crucial role in this decomposition.

5. How can idempotents be used to solve problems in the ring Z/100Z?

Idempotents can be used to simplify calculations in the ring Z/100Z. For example, if we need to find the product of two elements a and b in the ring, we can split the calculation into two parts by using the idempotents as coefficients. This can make the calculation more manageable and efficient.

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