The Math Forum

Search All of the Math Forum:

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

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

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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 ]

Posts: 12,067
Registered: 7/15/05
Re: Finite Rings
Posted: Feb 4, 2013 12:50 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

quasi wrote:
>Butch Malahide wrote:
>>quasi wrote:
>>> William Elliot wrote:

>>> >[In forum "Ask an Algebraist", user "Anu" asks:]
>>> >> 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?

>>> >No - the trivial ring.
>>> >So add the premise that R has a nonzero element.

>>> Even with that correction, the answer is still "no".
>>> Consider the commutative ring R consisting of the following
>>> seven distinct elements:
>>>    0, x, y, z, x+y, y+z, z+x

>>Does this mean that the additive group of R is a group of order 7?

>>> Besides the usual laws required for R to be a commutative
>>> ring (without identity), we also require the following
>>> relations:
>>>    r^2 = r for all r in R
>>>    r+r = 0 for all r in R

>>If r is nonzero, then r is an element of order 2 in the additive
>>group? What about Lagrange's theorem?

>Oops. I missed the element x+y+z.
>But then x+y+z is an annihilator -- which destroys my attempted

That's wrong too.

x+y+z is not an annihilator.

However it's not a zero-divisor -- in fact, it's an identity,

Thus, my example still fails.

>>>    xy = yz = zx = 0
>>> Note that the above relations imply
>>>    (x+y)z = (y+z)x = (z+x)y = 0
>>> so every element of R is a zerodivisor.
>>> However, since all elements of R are idempotent, it follows
>>> that no nonzero element of R annihilates all elements of R.


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

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.