Finiteness of a non-commutative ring

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Discussion Overview

The discussion revolves around the properties of non-commutative rings, specifically addressing whether a non-commutative ring with a finite number of non-units can be classified as a finite ring. The scope includes theoretical exploration of ring properties and potential counter-examples.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes that if R is a non-commutative ring with a finite number of non-units, it raises the question of whether R must be a finite ring.
  • Another participant mentions that the product of two rings, RxF, is a ring, and discusses the implications if R is a finite non-commutative ring and F is a field.
  • Subsequent replies suggest that RxF could be an infinite non-commutative ring, leading to further questioning about whether this serves as a counter-example to the initial proposition.
  • There is a correction from a participant who acknowledges a misunderstanding regarding the question posed.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the properties of RxF, with some suggesting it may serve as a counter-example while others challenge this interpretation. The discussion remains unresolved regarding the classification of R based on the finiteness of its non-units.

Contextual Notes

The discussion does not clarify the assumptions regarding the definitions of finite and infinite rings, nor does it resolve the mathematical implications of the examples provided.

Who May Find This Useful

Readers interested in abstract algebra, particularly in the properties of non-commutative rings and their classifications, may find this discussion relevant.

xixi
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Let R be a non-commutative ring . Suppose that the number of non-units of R is finite . Can we say that R is a finite ring?
 
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If R and F are rings, then RxF is a ring. In particular, if R is a finite non-commutative ring and F is a field...
 
zhentil said:
If R and F are rings, then RxF is a ring. In particular, if R is a finite non-commutative ring and F is a field...

so then??...
 
then RxF is an infinite non-commutative ring...
so...
 
Landau said:
then RxF is an infinite non-commutative ring...
so...

Do you mean that RxF is a counter-example ? but RxF has infinite non-unit elements .
 
Silly me! I should read the question.
 

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