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Generalization of sharp and core partial order using annihilators. (English) Zbl 1322.47037

Let \(R\) be a ring with unit \(1\). \(p \in R\) is called idempotent if \(p^2=p\). For \(a \in R\), let \({}^\circ a=\{x \in R: xa=0\}\) and \(a^{\circ} =\{x \in R: ax=0\}\). Let \({\mathcal I}_R=\{a \in R:{}^{\circ}a = {}^{\circ}p\) and \(a^{\circ} =p^{\circ}\) for some idempotent \(p \in R \}\). It follows that for any \(a \in {\mathcal I}_R\) such a \(p\) is unique and will be denoted by \(p_a\). The author defines the sharp order in \(R\) as follows: For \(a,b \in R\), we say that \(a {\leq}^{\sharp} b\) if \(a \in {\mathcal I}_R\) and \(a=p_ab=bp_a\). It is shown that the sharp order is indeed a partial order on \(R\). A characterization result is proved for two elements to be related by the sharp order. The core order in a ring \(R\) is also studied. These results extend some well-known results for operators on Banach spaces, obtained recently [M. A. Efimov, Math. Notes 93, No. 5, 784–788 (2013); translation from Mat. Zametki 93, No. 5, 794–797 (2013; Zbl 1303.47003)].

MSC:

47C10 Linear operators in \({}^*\)-algebras
06A06 Partial orders, general
15A09 Theory of matrix inversion and generalized inverses
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
16U99 Conditions on elements

Citations:

Zbl 1303.47003
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References:

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