Topic
Idempotence
About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.
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TL;DR: In this paper, the notion of a pair of complementary idempotents in a triangulated monoidal category, as well as more general IDempotent decompositions of identity, was introduced.
Abstract: In these notes we develop some basic theory of idempotents in monoidal categories. We introduce and study the notion of a pair of complementary idempotents in a triangulated monoidal category, as well as more general idempotent decompositions of identity. If $\mathbf{E}$ is a categorical idempotent then $\operatorname{End}(\mathbf{E})$ is a graded commutative algebra. The same is true of $\operatorname{Hom}(\mathbf{E},\mathbf{E}^c[1])$ under certain circumstances, where $\mathbf{E}^c$ is the complement. These generalize the notions of cohomology and Tate cohomology of a finite dimensional Hopf algebra, respectively.
13 citations
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TL;DR: In this article, the authors studied the subsets of a finite ring with identity that can be closed under multiplication and the implications that fact has to the structure of R. They achieved this by studying the properties of units that are preserved by idempotents.
13 citations
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TL;DR: In this paper, the well known Fitting decomposition is extended to the context of Jordan algebras, pairs, and triple systems, and it is shown that there exists a unique idempotent c of J such that a =a2 + a,,~ J2 @ Jo in the Peirce decomposition of J with respect to c, where a2 is invertible in J, and a, is nilpotent.
13 citations
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TL;DR: Nessary solvability conditions are derived using known results concerning eigenvectors of matrices in such structures based on linearly ordered commutative group where the role of ⊕ plays the maximum.
13 citations
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TL;DR: In this article, the sharp order is extended to the set of elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent.
Abstract: The sharp order is a well known partial order defined on the set of complex matrices with index less or equal one. Following Semrl's approach, Efimov extended this order to the set of those bounded Banach space operators $A$ for which the closure of the image and kernel are topologically complementary subspaces. In order to extend the sharp order to arbitrary ring $R$ (particulary to Rickart and Rickart $*$-rings) we use the notions of annihilators. The concept of the sharp order is extended to the set $\mathcal{I}_R$ of those elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent. It is proved that the sharp order is a partial order relation on $\mathcal{I}_R$. Following the idea we also extend and discuss the recently introduced concept of core partial order.
13 citations