In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely ...determined, then as an application, lifting idempotents modulo the Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is also characterized in terms of certain connected sets related to that ideal. Then as an application, we obtain that the sum of a lifting ideal and a regular ideal is a lifting ideal. We prove that lifting idempotents preserves the orthogonality in countable cases. The lifting property of an arbitrary morphism of rings is characterized. As another major result, it is proved that the number of idempotents of a ring
R
is finite if and only if it is of the form
2
κ
where
κ
is the cardinal of the connected components of
Spec
(
R
)
. Finally, it is proved that the primitive idempotents of a zero dimensional ring are in 1-1 correspondence with the isolated points of its prime spectrum. These results either generalize or improve several important results in the literature.
Surjective Lp-isometries on rank 1 idempotents Qian, Wenhua; Xiang, Zhang; Wu, Wenming ...
Journal of mathematical analysis and applications,
02/2023, Volume:
518, Issue:
1
Journal Article
Peer reviewed
We have studied surjective Lp-isometries on rank 1 idempotents acting on a Hilbert space H for p>1. We show that if φ is a surjective Lp-isometry on the set of rank 1 idempotents, then it leaves the ...set of rank 1 projections invariant. By applying the classical Wigner's theorem and some further calculations, we prove that either φ or φ⁎ is induced by a unitary or an anti-unitary.
Let n∈Z≥4 and Hq(Dn) be the semisimple Hecke algebra of type Dn with Hecke parameter q∈K×. For each simple Hq(Dn)-module V, we use the Hecke generators of Hq(Dn) to construct explicitly a ...quasi-idempotent zV (i.e., zV2=cVzV for some cV∈K×) which is defined over a natural integral form of Hq(Dn), such that eV:=cV−1zV is a primitive idempotent and eVHq(Dn)≅V as right Hq(Dn)-module. We use the seminormal bases of the Hecke algebra Hq(Bn) of type Bn to construct a complete set of pairwise orthogonal primitive idempotents of Hq(Dn), to obtain an explicit seminormal basis of Hq(Dn) as well as a new seminormal construction for each simple module over Hq(Dn). As byproducts, we discover some rational property of certain square-roots of quotients of γ-coefficients for Hq(Bn), which play a key role in the proof of the main results of the paper.
The article explores research findings akin to Amitsur’s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In ...most results related to rings and semirings, Birkenmeier’s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.
We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's ...relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated subsemigroup of each ideal; in particular, when an ideal is idempotent-generated, the rank and idempotent rank are equal. Along the way, we obtain a number of results of independent interest, and we demonstrate the utility of the semigroup theoretic approach by applying our results to obtain new proofs of some important representation theoretic results concerning the corresponding diagram algebras, the partial (or rook) Brauer algebra and Motzkin algebra.
We give a criterion for when idempotents of a ring R which commute modulo the Jacobson radical J(R) can be lifted to commuting idempotents of R. If such lifting is possible, we give extra information ...about the lifts. A “half-commuting” analogue is also proven, and this is used to give sufficient conditions for a ring to have the internal exchange property. In particular, we show that if R/J(R) is an internal exchange ring and idempotents lift modulo J(R), then R is an internal exchange ring. We also clarify some interesting results in the literature by investigating, and ultimately characterizing, the relationships between the finite (internal) exchange property, the (C3) property, and generalizations of square-free modules. We provide multiple examples delimiting these connections.
In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent ...commutator has a non-trivial common closed invariant subspace. We also present a geometric characterization of invariant subspaces of idempotents and classify operators that are essentially idempotent.
Idempotent linear relations Arias, M. Laura; Contino, Maximiliano; Maestripieri, Alejandra ...
Journal of mathematical analysis and applications,
12/2022, Volume:
516, Issue:
2
Journal Article
Peer reviewed
Open access
A linear relation E acting on a Hilbert space is idempotent if E2=E. A triplet of subspaces is needed to characterize a given idempotent: (ranE,ran(I−E),domE), or equivalently, ...(ker(I−E),kerE,mulE). The relations satisfying the inclusions E2⊆E (sub-idempotent) or E⊆E2 (super-idempotent) play an important role. Lastly, the adjoint and the closure of an idempotent linear relation are studied.
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