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Nilpotent elements in a ring

  • #1
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Homework Statement


Show that 0 is the only in R if and only if a^2 = 0 implies a = 0.

Homework Equations


none

The Attempt at a Solution


So I'm not sure if I'm doing this right.
a^2 = a*a = 0. Therefore, either a or a is zero.

The reason i'm not sure about this is because i'm thinking about matrices, where matrix A^2 can equal zero while A doesn't equal zero.

Also, did the logic that I use only work if the original question considered the ring a domain?
 

Answers and Replies

  • #2
RUber
Homework Helper
1,687
344
For the if and only if, you should have to demonstrate the proof both ways. If there is a unique zero, then ##a^2=0 \implies a=0##, and if ##a^2=0 \implies a=0##, then zero is unique.
 

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