Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order.

Wotzdorfg 2021-02-11 Answered
Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Ayesha Gomez
Answered 2021-02-12 Author has 104 answers

Consider p and q be any nonzero element of R.
therefore,
pq+pq+pq+...n×=n(pq)
=(n×p)×q
=p(n×q)
Since,
n×p=0
(n×p)q=0
p(n×q)=0
With no zero divisiors,
n×q=0, therefore,
and if the pq+pq+pq+...m×=m(pq)
=(mp)q
=p(mq)
and, mq=0
(mp)q=0
p(mq)=0
with no zero divisors, , mq=0
Therefore, nmandmn

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question