A Von Neumann ring

Groups, Rings, Domains, Modules, etc, Galois theory
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Tolaso J Kos
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A Von Neumann ring

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Post by Tolaso J Kos »

Let $(\mathcal{R}, +, \cdot)$ be a ring without necessary a unitary element but it has at least two elements. If $\mathcal{R}$ furthermore satisfies the property: "for every element $a \in \mathcal{R}$ that is not zero ($a \neq 0$) there exists a unique element $b \in \mathcal{R}$ such that $aba=a$ , then prove that:
  1. $\mathcal{R}$ does not have zero divisors.
  2. $bab=b$.
  3. $\mathcal{R}$ has a unitary element.
  4. $\mathcal{R}$ is a divisor ring.
Imagination is much more important than knowledge.

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