All Questions
4 questions
6
votes
1
answer
256
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Is there any structural characterization of the rings in which every element other than the identity is a (two-sided) zero divisor?
[I fear that I'm missing something obvious here, but I'll dare to ask anyway.]
As we all know, a division ring is a (unital, associative, non-zero) ring where every non-zero element is a unit. So, let ...
- 10.5k
21
votes
1
answer
2k
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Is there any non-commutative ring such that every element other than the identity is a zero divisor?
A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring"...
- 10.5k
2
votes
1
answer
192
views
Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR
Let $H$ be a (commutative or non-commutative) monoid. We say that $H$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ...
- 10.5k
2
votes
0
answers
98
views
Do $r(a) \leq^\oplus R$ and $r(a) = r(a^2)$ imply $r(a) = eR$ and $aR \subseteq (1-e)R$ for some idempotent $e$?
Let $R$ be a (commutative or non-commutative, associative) ring with unity, and let $a$ be an element of $R$ such that $r(a) = r(a^2)$, where $r(\cdot)$ denotes a right annihilator. It follows that $r(...
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