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Taheri Zahra - One of the best experts on this subject based on the ideXlab platform.

  • Stone type representations and dualities by power set Ring
    2021
    Co-Authors: Tarizadeh Abolfazl, Taheri Zahra
    Abstract:

    In this paper, it is shown that the Boolean Ring of a commutative Ring is isomorphic to the Ring of clopens of its prime spectrum. In particular, Stone's Representation Theorem is generalized. The prime spectrum of the Boolean Ring of a given Ring $R$ is identified with the Pierce spectrum of $R$. The discreteness of prime spectra is characterized. It is also proved that the space of connected components of a compact space $X$ is isomorphic to the prime spectrum of the Ring of clopens of $X$. As another major result, it is shown that a morphism of Rings between complete Boolean Rings preserves suprema if and only if the induced map between the corresponding prime spectra is an open map.Comment: 19 page

  • Stone type representation theorems and dualities by power set Ring
    2020
    Co-Authors: Tarizadeh Abolfazl, Taheri Zahra
    Abstract:

    In this paper, it is shown that the Boolean Ring of a commutative Ring is isomorphic to the clopen Ring of its prime spectrum. The topological version of this result is also proved which states that the space of connected components of a compact space is homeomorphic to the prime spectrum of its clopens. In particular, Stone's Representation Theorem is generalized from Boolean Rings to arbitrary commutative Rings and also the Stone duality is easily deduced. The prime spectrum of the Boolean Ring of a given Ring $R$ is identified with the Pierce spectrum of $R$. Then as an application, it is shown that the prime spectrum of a Ring $R$ is discrete iff the Boolean Ring of $R$ is isomorphic to the power set Ring of its prime spectrum. As another major result, it is shown that a morphism of Rings between complete Boolean Rings preserves suprema iff the induced map between the corresponding prime spectra is an open map. This result leads us to a Stone type duality which states that the category of complete Boolean Rings is the dual of the category of compact extremally disconnected spaces. This duality in particular yields that the injective objects of the category of Boolean Rings are precisely the complete Boolean Rings. New characterizations for the completeness of Boolean Rings are also given. Finally, general results in the fixed-point theory have been obtained.Comment: 19 page

Tarizadeh Abolfazl - One of the best experts on this subject based on the ideXlab platform.

  • Stone type representations and dualities by power set Ring
    2021
    Co-Authors: Tarizadeh Abolfazl, Taheri Zahra
    Abstract:

    In this paper, it is shown that the Boolean Ring of a commutative Ring is isomorphic to the Ring of clopens of its prime spectrum. In particular, Stone's Representation Theorem is generalized. The prime spectrum of the Boolean Ring of a given Ring $R$ is identified with the Pierce spectrum of $R$. The discreteness of prime spectra is characterized. It is also proved that the space of connected components of a compact space $X$ is isomorphic to the prime spectrum of the Ring of clopens of $X$. As another major result, it is shown that a morphism of Rings between complete Boolean Rings preserves suprema if and only if the induced map between the corresponding prime spectra is an open map.Comment: 19 page

  • Stone type representation theorems and dualities by power set Ring
    2020
    Co-Authors: Tarizadeh Abolfazl, Taheri Zahra
    Abstract:

    In this paper, it is shown that the Boolean Ring of a commutative Ring is isomorphic to the clopen Ring of its prime spectrum. The topological version of this result is also proved which states that the space of connected components of a compact space is homeomorphic to the prime spectrum of its clopens. In particular, Stone's Representation Theorem is generalized from Boolean Rings to arbitrary commutative Rings and also the Stone duality is easily deduced. The prime spectrum of the Boolean Ring of a given Ring $R$ is identified with the Pierce spectrum of $R$. Then as an application, it is shown that the prime spectrum of a Ring $R$ is discrete iff the Boolean Ring of $R$ is isomorphic to the power set Ring of its prime spectrum. As another major result, it is shown that a morphism of Rings between complete Boolean Rings preserves suprema iff the induced map between the corresponding prime spectra is an open map. This result leads us to a Stone type duality which states that the category of complete Boolean Rings is the dual of the category of compact extremally disconnected spaces. This duality in particular yields that the injective objects of the category of Boolean Rings are precisely the complete Boolean Rings. New characterizations for the completeness of Boolean Rings are also given. Finally, general results in the fixed-point theory have been obtained.Comment: 19 page

Chen Huanyin - One of the best experts on this subject based on the ideXlab platform.

  • On Medium *-Clean Rings
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Chen Huanyin, Abdolyousefi, Marjan Sheibani, Kose Handan
    Abstract:

    WOS: 000454919400015A *-Ring R is called a medium *-clean Ring if every element in R is the sum or difference of an element in its Jacobson radical and a projection that commute. We prove that a Ring R is medium *-clean if and only if R is strongly *-clean and R/J(R) is a Boolean Ring, Z3 or the product of such Rings, if and only if R weakly J-*-clean and a2R is uniquely *-clean for all aR, if and only if every idempotent lifts modulo J(R), R is abelian and R/J(R) weakly *-Boolean. A subclass of medium *-clean Rings with many nilpotents is thereby characterized.Natural Science Foundation of Zhejiang Province, ChinaNatural Science Foundation of Zhejiang Province [LY17A010018]The authors would like to thank the referee for his/her careful reading and valuable remarks that improved the presentation of our work. H. Chen was supported by the Natural Science Foundation of Zhejiang Province, China (no. LY17A010018)

  • Strongly 2-nil-clean Rings with involutions
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Chen Huanyin, Sheibani Abdolyousefi Marjan
    Abstract:

    summary:A $*$-Ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-Rings are obtained. We prove that a $*$-Ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN Ring, if and only if $R$ is abelian, $J(R)$ is nil and $R/J(R)$ is $*$-tripotent. Furthermore, we explore the structure of such Rings and prove that a $*$-Ring $R$ is strongly 2-nil-$*$-clean if and only if $R$ is abelian and $R\cong R_1, R_2$ or $R_1\times R_2$, where $R_1/J(R_1)$ is a $*$-Boolean Ring and $J(R_1)$ is nil, $R_2/J(R_2)$ is a $*$-Yaqub Ring and $J(R_2)$ is nil. The uniqueness of projections of such Rings are thereby investigated

  • Rings consisting entirely of certain elements
    'Rocky Mountain Mathematics Consortium', 2018
    Co-Authors: Chen Huanyin, Sheibani Marjan, Ashrafi Nahid
    Abstract:

    summary:We completely determine when a Ring consists entirely of weak idempotents, units and nilpotents. We prove that such Ring is exactly isomorphic to one of the following: a Boolean Ring; $\Bbb Z_3\oplus {\Bbb Z}_3$; $\Bbb Z_3\oplus B$ where $B$ is a Boolean Ring; local Ring with nil Jacobson radical; $M_2(\Bbb Z_2)$ or $M_2(\Bbb Z_3)$; or the Ring of a Morita context with zero paiRings where the underlying Rings are $\Bbb Z_2$ or $\Bbb Z_3$

  • Certain decompositions of matrices over Abelian Rings
    'Rocky Mountain Mathematics Consortium', 2017
    Co-Authors: Ashrafi Nahid, Sheibani Marjan, Chen Huanyin
    Abstract:

    summary:A Ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix Rings over abelian Rings. Let $R$ be abelian, and let $n\in {\Bbb N}$. We prove that $M_n(R)$ is nil clean if and only if $R/J(R)$ is Boolean and $M_n(J(R))$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R/J(R)$ is ${\Bbb Z}_3$, $B$ or ${\Bbb Z}_3\oplus B$ where $B$ is a Boolean Ring, and that $M_n(R)$ is weakly nil clean if and only if $M_n(R)$ is nil clean for all $n\geq 2$

Vourdas A. - One of the best experts on this subject based on the ideXlab platform.

  • Coherent spaces, Boolean Rings and quantum gates
    'Elsevier BV', 2016
    Co-Authors: Vourdas A.
    Abstract:

    Coherent spaces spanned by a finite number of coherent states, are introduced. Their coherence properties are studied, using the Dirac contour representation. It is shown that the corresponding projectors resolve the identity, and that they transform into projectors of the same type, under displacement transformations, and also under time evolution. The set of these spaces, with the logical OR and AND operations is a distributive lattice, and with the logical XOR and AND operations is a Boolean Ring (Stone\rq{}s formalism). Applications of this Boolean Ring into classical CNOT gates with $n$-ary variables, and also quantum CNOT gates with coherent states, are discussed

Vourdas Apostolos - One of the best experts on this subject based on the ideXlab platform.

  • Coherent spaces, Boolean Rings and quantum gates
    'Elsevier BV', 2016
    Co-Authors: Vourdas Apostolos
    Abstract:

    YesCoherent spaces spanned by a nite number of coherent states, are introduced. Their coherence properties are studied, using the Dirac contour representation. It is shown that the corresponding projectors resolve the identity, and that they transform into projectors of the same type, under displacement transformations, and also under time evolution. The set of these spaces, with the logical OR and AND operations is a distributive lattice, and with the logical XOR and AND operations is a Boolean Ring (Stone's formalism). Applications of this Boolean Ring into classical CNOT gates with n-ary variables, and also quantum CNOT gates with coherent states, are discussed