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On \(E_ k\)-rings. (English) Zbl 0673.16009

Let \(R\) be a non-Boolean ring, let \(E\) denote its set of idempotents, and assume \(| E| >1\). For a given positive integer \(k\), call \(R\) an \(E_ k\)-ring if every subring of \(R\) containing \(k\) nonzero idempotents either contains \(E\) or is contained in \(E\). If in addition \(| E| =k+1\), then call \(R\) trivial.
The principal results are as follows: (I) \(R\) is an \(E_ 1\)-ring if and only if \(E\) is a proper subset of \(R\) and \(| E| =2\). - (II) \(R\) is a nontrivial \(E_ 2\)-ring if and only if \(E\) is a proper subsemigroup of \(R\) such that one of the following holds: (i) \(E\) is commutative with identity and \(| E| =4\). (ii) \(E\setminus \{0\}\) is a left (right) zero semigroup of prime order \(p>2\) and there exist \(e\in E\setminus \{0\}\) and \(a\in R\setminus \{0\}\) such that \(E\setminus \{0\}=\{e+ka|\) \(k=0,1,...,p-1\}\). - (III) \(R\) is a nontrivial regular \(E_ 2\)-ring if and only if it is the direct sum of two division rings and \(| R| >4.\)
{There are two misprints in statements of results: in Lemma 1.1, “\(2e=0\)” should replace “\(Re=0\)”, and in Theorem 2.3, \(N\) denotes the set of nilpotent elements.}
Reviewer: H.E.Bell

MSC:

16U99 Conditions on elements
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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References:

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