Is Idempotent Equivalence in Rings Transitive?

In summary, two idempotents P and Q in a ring are considered equivalent if there exist elements X and Y such that P = XY and Q = YX. The transitive property of this relation can be proven by showing that if P~Q and Q~R, then P~R. This can be done by showing that P = XY and R = WV, and therefore P~R. This proof may seem complicated, but understanding the properties of idempotents can help in understanding it.
  • #1
cogito²
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One of my books defines a relation which is "evidently" an equivalence relation. It says that two idempotents in a ring P and Q are said to be equivalent if there exist elements X and Y such that P = XY and Q = YX.

The proof that this relation is transitive eludes me. There is so little information, that I feel like this has to have a really short proof, but I just can't seem to figure it out (or find it on the magical internet). If anyone can can ease my frustration, I would be grateful.
 
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  • #2
If P~Q and Q~R then, P=XY, Q=YX and Q=VW, R=WV.
Then,
P=P2=XYXY=XQY=(XV)(WY)
R=R2=WVWV=WQV=(WY)(XV)
so P~R.
 
  • #3
Alright that's about as complicated as I expected it to be...I basically had that written down, but apparently I don't quite have a fully functioning brain and for some reason couldn't see it. Many thanks.
 

FAQ: Is Idempotent Equivalence in Rings Transitive?

What is idempotent equivalence in rings?

Idempotent equivalence in rings is a mathematical concept that refers to the relationship between two elements in a ring that have the same value when raised to a certain power. In other words, two elements are idempotent equivalent if they have the same value when multiplied by themselves a certain number of times.

How is idempotent equivalence different from equality in rings?

While equality in rings requires two elements to have the exact same value, idempotent equivalence allows for some variation in the value as long as it is the same when raised to a certain power. This means that idempotent equivalence is a weaker form of equality.

What is the significance of idempotent equivalence in ring theory?

Idempotent equivalence plays an important role in the study of rings and other algebraic structures. It helps to identify elements with similar properties and allows for simplification of complex calculations. It also has applications in areas such as coding theory and cryptography.

Can two elements be idempotent equivalent in one ring but not in another?

Yes, two elements can be idempotent equivalent in one ring but not in another. This is because the concept of idempotent equivalence is dependent on the specific ring and the operations defined within it. Different rings may have different rules for determining idempotent equivalence.

Are there any practical applications of idempotent equivalence in real-world scenarios?

Yes, idempotent equivalence has practical applications in various fields such as computer science, economics, and physics. For example, it is used in the field of coding theory to design error-correcting codes and in economics to study economic systems. It also has applications in physics for studying symmetry and conservation laws.

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