Two questions:

1. If R is a ring then is every element of R either a unit or zero-divisor.

COnversely

2. IF R is a ring, then there is no elements of R which is both a unit and a zero-divisor.

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- Mar 4th 2007, 12:02 PMchadlyterRing help
Two questions:

1. If R is a ring then is every element of R either a unit or zero-divisor.

COnversely

2. IF R is a ring, then there is no elements of R which is both a unit and a zero-divisor. - Mar 4th 2007, 04:44 PMThePerfectHacker
Is R a finite ring?

Because, if it is then we can prove it as follows:

If x is a zero-divisor the proof is complete.

If x is not a zero-divisor then consider the finite set,

{a_1x,a_2x,...,a_nx} where a_i are the non-zero elements of the ring (assuming it is non-trivial).

Then, we can show none are equal to each other.

(Excercise).

Then by Dirichlet's Pigeonhole Principle, is an enumeration of all non-zero elements in R. Hence we can find an a_i such that a_i x = 1. Thus, x is a unit. - Mar 4th 2007, 07:32 PMThePerfectHackerQuote:

Originally Posted by**TheNextDirichlet**

Because, for example, consider, Z.

The integers under addition and multiplication.

The element 2 in Z, it neither a unit nor zero divisor :eek:

(However, for all finite rings this is true). - Mar 5th 2007, 06:18 AMchadlyterRIng
The Ring is not defined as a finite ring. These are true/false questions I have encountered in a abstract algebra class. If false I have to give a counterexample. For example, If R is a ring, then every element of R is either a unit or zero-divisor. In my mind this is false but the counterexample is what is messing with my head.

- Mar 5th 2007, 07:02 AMThePerfectHacker