Drexel dragonThe Math ForumDonate to the Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Topic: Finite Rings
Replies: 28   Last Post: Feb 6, 2013 8:33 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
William Elliot

Posts: 2,383
Registered: 1/8/12
Re: Finite Rings
Posted: Feb 5, 2013 4:12 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Sun, 3 Feb 2013, William Elliot wrote:

> > If R is a finite commutative ring without multiplicative identity
> > and if every element is a zero divisor, then does there exist
> > a nonzero element which annihilates all elements of the ring?

Assume without debate nor dispute, that R has a non-zero element p.

If there's no nonzero annihilator, then for all nonzero x,
there's some a_x with nonzero a a_x.

Let p0 = p and uning induction, for all j in N, define
p_(j+1) = pj a_pj.
p0, p1, p2, ... is sequence of nonzero elements.

Since R is finite, there's some distinct j,k with pj = pk.
Thusly we've some nonzero a,b with ab = a. Now what?

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2015. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.