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V V Bavula - One of the best experts on this subject based on the ideXlab platform.
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criteria for a Ring to have aleft noetherian largest left Quotient Ring
Algebras and Representation Theory, 2018Co-Authors: V V BavulaAbstract:Criteria are given for a Ring to have a left Noetherian largest left Quotient Ring. It is proved that each such a Ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian Ring has only finitely many maximal left denominator sets.
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criteria for a Ring to have a left noetherian left Quotient Ring
Journal of Pure and Applied Algebra, 2017Co-Authors: V V BavulaAbstract:Two criteria are given for a Ring to have a left Noetherian left Quotient Ring (to find a criterion was an open problem since 70's). It is proved that each such Ring has only finitely many maximal left denominator sets.
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criteria for a Ring to have a left noetherian left Quotient Ring
arXiv: Rings and Algebras, 2015Co-Authors: V V BavulaAbstract:Two criteria are given for a Ring to have a left Noetherian left Quotient Ring (this was an open problem since 70's). It is proved that each such Ring has only finitely many maximal left denominator sets.
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new criteria for a Ring to have a semisimple left Quotient Ring
Journal of Algebra and Its Applications, 2015Co-Authors: V V BavulaAbstract:Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a Ring to have a semisimple left Quotient Ring. The aim of the paper is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary Ring R the set ℳ of maximal left denominator sets of R is a non-empty set [V. V. Bavula, The largest left Quotient Ring of a Ring, preprint (2011), arXiv:math.RA:1101.5107]: Theorem (The First Criterion). A Ring R has a semisimple left Quotient Ring Q iff ℳ is a finite set, ⋂S∈ℳ ass (S) = 0 and, for each S ∈ ℳ, the Ring S-1R is a simple left Artinian Ring. In this case, Q ≃ ∏S∈ℳ S-1R. The Second Criterion is given via the minimal primes of R and goes further than the First one in the sense that it describes explicitly the maximal left denominator sets S via the minimal primes of R. The Third Criterion is close to Goldie's Criterion but it is easier to check in applications (basically, it reduces Goldie's Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets.
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the algebra of polynomial integro differential operators is a holonomic bimodule over the subalgebra of polynomial differential operators
Algebras and Representation Theory, 2014Co-Authors: V V BavulaAbstract:In contrast to its subalgebra \(A_n:=K\langle x_1, \ldots , x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}\rangle \) of polynomial differential operators (i.e. the n’th Weyl algebra), the algebra \({\mathbb{I}}_n:=K\langle x_1, \ldots ,\)\( x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}, \int_1, \ldots , \int_n\rangle \) of polynomial integro-differential operators is neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that \({\mathbb{I}}_n\) is a left (right) coherent algebra iff n = 1; the algebra \({\mathbb{I}}_n\) is a holonomic An-bimodule of length 3n and has multiplicity 3n with respect to the filtration of Bernstein, and all 3n simple factors of \({\mathbb{I}}_n\) are pairwise non-isomorphic An-bimodules. The socle length of the An-bimodule \({\mathbb{I}}_n\) is n + 1, the socle filtration is found, and the m’th term of the socle filtration has length \({n\choose m}2^{n-m}\). This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra \({\mathbb{I}}_n\) is the maximal left (resp. right) order in the largest left (resp. right) Quotient Ring of the algebra \({\mathbb{I}}_n\).
Bavula V.v. - One of the best experts on this subject based on the ideXlab platform.
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Left localizable Rings and their characterizations
'Elsevier BV', 2019Co-Authors: Bavula V.v.Abstract:A new class of Rings, the class of left localizable Rings, is introduced. A Ring R is left localizable if each nonzero element of R is invertible in some left localization S-1R of the Ring R. Explicit criteria are given for a Ring to be a left localizable Ring provided the Ring has only finitely many maximal left denominator sets (e.g., this is the case if a Ring has a left Noetherian left Quotient Ring). It is proved that a Ring with finitely many maximal left denominator sets is a left localizable Ring iff its left Quotient Ring is a direct product of finitely many division Rings. A characterization is given of the class of Rings that are finite direct product of left localization maximal Rings
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The classical left regular left Quotient Ring of a Ring and its semisimplicity criteria
'World Scientific Pub Co Pte Lt', 2016Co-Authors: Bavula V.v.Abstract:Let R be a Ring, CR and ′ CR be the set of regular and left regular elements of R (CR ⊆ ′ CR). Goldie’s Theorem is a semisimplicity criterion for the classical left Quotient Ring Ql,cl(R) := C −1 R R. Semisimplicity criteria are given for the classical left regular left Quotient Ring ′Ql,cl(R) := ′ C −1 R R. As a corollary, two new semisimplicity criteria for Ql,cl(R) are obtained (in the spirit of Goldie)
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Weakly left localizable Rings
'Informa UK Limited', 2016Co-Authors: Bavula V.v.Abstract:A new class of Rings, the class of weakly left localizable Rings, is introduced. A Ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S−1R of the Ring R. Explicit criteria are given for a Ring to be a weakly left localizable Ring provided the Ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian Rings). It is proved that a Ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable Ring iff its left Quotient Ring is a direct product of finitely many local Rings such that their radicals are nil ideals
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The classical left regular left Quotient Ring of a Ring and its semisimplicity criteria
2015Co-Authors: Bavula V.v.Abstract:Let $R$ be a Ring, $\CC_R$ and $\pCCR$ be the set of regular and left regular elements of $R$ ($\CC_R\subseteq \pCCR$). Goldie's Theorem is a semisimplicity criterion for the classical left Quotient Ring $Q_{l,cl}(R):=\CC_R^{-1}R$. Semisimplicity criteria are given for the classical left regular left Quotient Ring $'Q_{l,cl}(R):=\pCCR^{-1}R$. As a corollary, two new semisimplicity criteria for $Q_{l,cl}(R)$ are obtained (in the spirit of Goldie).Comment: 22 page
Lizhen Zhang - One of the best experts on this subject based on the ideXlab platform.
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erd h o s burgess constant of the multiplicative semigroup of the Quotient Ring of mathbb f _q x
arXiv: Combinatorics, 2018Co-Authors: Jun Hao, Haoli Wang, Lizhen ZhangAbstract:Let $\mathcal{S}$ be a semigroup endowed with a binary associative operation $*$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e*e=e$. The {\sl Erd\H{o}s-Burgess constant} of the semigroup $\mathcal{S}$ is defined as the smallest $\ell\in \mathbb{N}\cup \{\infty\}$ such that any sequence $T$ of terms from $S$ and of length $\ell$ contains a nonempty subsequence the product of whose terms, in some order, is idempotent. Let $q$ be a prime power, and let $\F_q[x]$ be the Ring of polynomials over the finite field $\F_q$. Let $R=\F_q[x]\diagup K$ be a Quotient Ring of $\F_q[x]$ modulo any ideal $K$. We gave a sharp lower bound of the Erd\H{o}s-Burgess constant of the multiplicative semigroup of the Ring $R$, in particular, we determined the Erd\H{o}s-Burgess constant in the case when $K$ is factored into either a power of some prime ideal or a product of some pairwise distinct prime ideals in $\F_q[x]$.
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on the erd h o s burgess constant of the multiplicative semigroup of a factor Ring of mathbb f _q x
arXiv: Combinatorics, 2018Co-Authors: Haoli Wang, Jun Hao, Lizhen ZhangAbstract:Let $\mathcal{S}$ be a commutative semigroup endowed with a binary associative operation $+$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The {\sl Erdős-Burgess constant} of $\mathcal{S}$ is defined as the smallest $\ell\in \mathbb{N}\cup \{\infty\}$ such that any sequence $T$ of terms from $S$ and of length $\ell$ contains a nonempty subsequence the sum of whose terms is idempotent. Let $q$ be a prime power, and let $\F_q[x]$ be the polynomial Ring over the finite field $\F_q$. Let $R=\F_q[x]\diagup K$ be a Quotient Ring of $\F_q[x]$ modulo any ideal $K$. We gave a sharp lower bound of the Erdős-Burgess constant of the multiplicative semigroup of the Ring $R$, in particular, we determined the Erdős-Burgess constant in the case when $K$ is the power of a prime ideal or a product of pairwise distinct prime ideals in $\F_q[x]$.
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note on the davenport constant of the multiplicative semigroup of the Quotient Ring x1d53d p x f x
International Journal of Number Theory, 2016Co-Authors: Haoli Wang, Lizhen Zhang, Qinghong WangAbstract:Let 𝒮 be a finite commutative semigroup. The Davenport constant of 𝒮, denoted D(𝒮), is defined to be the least positive integer l such that every sequence T of elements in 𝒮 of length at least l contains a proper subsequence T′ with the sum of all terms from T′ equaling the sum of all terms from T. Let 𝔽p[x] be a polynomial Ring in one variable over the prime field 𝔽p, and let f(x) ∈ 𝔽p[x]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the Quotient Ring 𝔽p[x] 〈f(x)〉 and proved that, for any prime p > 2 and any polynomial f(x) ∈ 𝔽p[x] which factors into a product of pairwise non-associate irreducible polynomials, D(𝒮f(x)p) = D(U(𝒮 f(x)p)), where 𝒮f(x)p denotes the multiplicative semigroup of the Quotient Ring 𝔽p[x] 〈f(x)〉 and U(𝒮f(x)p) denotes the group of units of the semigroup 𝒮f(x)p.
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a problem of wang on davenport constant for the multiplicative semigroup of the Quotient Ring of f_2 x
arXiv: Combinatorics, 2015Co-Authors: Lizhen Zhang, Haoli WangAbstract:Let $\F_q[x]$ be the Ring of polynomials over the finite field $\F_q$, and let $f$ be a polynomial of $\F_q[x]$. Let $R=\frac{\F_q[x]}{(f)}$ be a Quotient Ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. Let $\mathcal{S}_R$ be the multiplicative semigroup of the Ring $R$, and let ${\rm U}(\mathcal{S}_R)$ be the group of units of $\mathcal{S}_R$. The Davenport constant ${\rm D}(\mathcal{S}_R)$ of the multiplicative semigroup $\mathcal{S}_R$ is the least positive integer $\ell$ such that for any $\ell$ polynomials $g_1,g_2,\ldots,g_{\ell}\in \F_q[x]$, there exists a subset $I\subsetneq [1,\ell]$ with $$\prod\limits_{i\in I} g_i \equiv \prod\limits_{i=1}^{\ell} g_i\pmod f.$$ In this manuscript, we proved that for the case of $q=2$, $${\rm D}({\rm U}(\mathcal{S}_R))\leq {\rm D}(\mathcal{S}_R)\leq {\rm D}({\rm U}(\mathcal{S}_R))+\delta_f,$$ where \begin{displaymath} \delta_f=\left\{\begin{array}{ll} 0 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),\ f)=1_{\F_{2}}$}\\ 1 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),\ f)\in \{x, \ x+1_{\mathbb{F}_2}\}$}\\ 2 & \textrm{if $gcd(x*(x+1_{\mathbb{F}_2}),f)=x*(x+1_{\mathbb{F}_2}) $}\\ \end{array} \right. \end{displaymath} which partially answered an open problem of Wang on Davenport constant for the multiplicative semigroup of $\frac{\F_q[x]}{(f)}$ (G.Q. Wang, \emph{Davenport constant for semigroups II,} Journal of Number Theory, 155 (2015) 124--134).
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davenport constant of the multiplicative semigroup of the Quotient Ring frac f_p x langle f x rangle
arXiv: Number Theory, 2014Co-Authors: Haoli Wang, Lizhen Zhang, Qinghong WangAbstract:Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial Ring in one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the Quotient Ring $\frac{\F_p[x]}{\langle f(x)\rangle}$. Among other results, we mainly prove that, for any prime $p>2$ and any polynomial $f(x)\in \F_p[x]$ which can be factorized into several pairwise non-associted irreducible polynomials in $\F_p[x]$, then $$D(\mathcal{S}_{f(x)}^p)=D(U(\mathcal{S}_{f(x)}^p)),$$ where $\mathcal{S}_{f(x)}^p$ denotes the multiplicative semigroup of the Quotient Ring $\frac{\F_p[x]}{\langle f(x)\rangle}$ and $U(\mathcal{S}_{f(x)}^p)$ denotes the group of units of the semigroup $\mathcal{S}_{f(x)}^p$.
Haoli Wang - One of the best experts on this subject based on the ideXlab platform.
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erd h o s burgess constant of the multiplicative semigroup of the Quotient Ring of mathbb f _q x
arXiv: Combinatorics, 2018Co-Authors: Jun Hao, Haoli Wang, Lizhen ZhangAbstract:Let $\mathcal{S}$ be a semigroup endowed with a binary associative operation $*$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e*e=e$. The {\sl Erd\H{o}s-Burgess constant} of the semigroup $\mathcal{S}$ is defined as the smallest $\ell\in \mathbb{N}\cup \{\infty\}$ such that any sequence $T$ of terms from $S$ and of length $\ell$ contains a nonempty subsequence the product of whose terms, in some order, is idempotent. Let $q$ be a prime power, and let $\F_q[x]$ be the Ring of polynomials over the finite field $\F_q$. Let $R=\F_q[x]\diagup K$ be a Quotient Ring of $\F_q[x]$ modulo any ideal $K$. We gave a sharp lower bound of the Erd\H{o}s-Burgess constant of the multiplicative semigroup of the Ring $R$, in particular, we determined the Erd\H{o}s-Burgess constant in the case when $K$ is factored into either a power of some prime ideal or a product of some pairwise distinct prime ideals in $\F_q[x]$.
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on the erd h o s burgess constant of the multiplicative semigroup of a factor Ring of mathbb f _q x
arXiv: Combinatorics, 2018Co-Authors: Haoli Wang, Jun Hao, Lizhen ZhangAbstract:Let $\mathcal{S}$ be a commutative semigroup endowed with a binary associative operation $+$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The {\sl Erdős-Burgess constant} of $\mathcal{S}$ is defined as the smallest $\ell\in \mathbb{N}\cup \{\infty\}$ such that any sequence $T$ of terms from $S$ and of length $\ell$ contains a nonempty subsequence the sum of whose terms is idempotent. Let $q$ be a prime power, and let $\F_q[x]$ be the polynomial Ring over the finite field $\F_q$. Let $R=\F_q[x]\diagup K$ be a Quotient Ring of $\F_q[x]$ modulo any ideal $K$. We gave a sharp lower bound of the Erdős-Burgess constant of the multiplicative semigroup of the Ring $R$, in particular, we determined the Erdős-Burgess constant in the case when $K$ is the power of a prime ideal or a product of pairwise distinct prime ideals in $\F_q[x]$.
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note on the davenport constant of the multiplicative semigroup of the Quotient Ring x1d53d p x f x
International Journal of Number Theory, 2016Co-Authors: Haoli Wang, Lizhen Zhang, Qinghong WangAbstract:Let 𝒮 be a finite commutative semigroup. The Davenport constant of 𝒮, denoted D(𝒮), is defined to be the least positive integer l such that every sequence T of elements in 𝒮 of length at least l contains a proper subsequence T′ with the sum of all terms from T′ equaling the sum of all terms from T. Let 𝔽p[x] be a polynomial Ring in one variable over the prime field 𝔽p, and let f(x) ∈ 𝔽p[x]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the Quotient Ring 𝔽p[x] 〈f(x)〉 and proved that, for any prime p > 2 and any polynomial f(x) ∈ 𝔽p[x] which factors into a product of pairwise non-associate irreducible polynomials, D(𝒮f(x)p) = D(U(𝒮 f(x)p)), where 𝒮f(x)p denotes the multiplicative semigroup of the Quotient Ring 𝔽p[x] 〈f(x)〉 and U(𝒮f(x)p) denotes the group of units of the semigroup 𝒮f(x)p.
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a problem of wang on davenport constant for the multiplicative semigroup of the Quotient Ring of f_2 x
arXiv: Combinatorics, 2015Co-Authors: Lizhen Zhang, Haoli WangAbstract:Let $\F_q[x]$ be the Ring of polynomials over the finite field $\F_q$, and let $f$ be a polynomial of $\F_q[x]$. Let $R=\frac{\F_q[x]}{(f)}$ be a Quotient Ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. Let $\mathcal{S}_R$ be the multiplicative semigroup of the Ring $R$, and let ${\rm U}(\mathcal{S}_R)$ be the group of units of $\mathcal{S}_R$. The Davenport constant ${\rm D}(\mathcal{S}_R)$ of the multiplicative semigroup $\mathcal{S}_R$ is the least positive integer $\ell$ such that for any $\ell$ polynomials $g_1,g_2,\ldots,g_{\ell}\in \F_q[x]$, there exists a subset $I\subsetneq [1,\ell]$ with $$\prod\limits_{i\in I} g_i \equiv \prod\limits_{i=1}^{\ell} g_i\pmod f.$$ In this manuscript, we proved that for the case of $q=2$, $${\rm D}({\rm U}(\mathcal{S}_R))\leq {\rm D}(\mathcal{S}_R)\leq {\rm D}({\rm U}(\mathcal{S}_R))+\delta_f,$$ where \begin{displaymath} \delta_f=\left\{\begin{array}{ll} 0 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),\ f)=1_{\F_{2}}$}\\ 1 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),\ f)\in \{x, \ x+1_{\mathbb{F}_2}\}$}\\ 2 & \textrm{if $gcd(x*(x+1_{\mathbb{F}_2}),f)=x*(x+1_{\mathbb{F}_2}) $}\\ \end{array} \right. \end{displaymath} which partially answered an open problem of Wang on Davenport constant for the multiplicative semigroup of $\frac{\F_q[x]}{(f)}$ (G.Q. Wang, \emph{Davenport constant for semigroups II,} Journal of Number Theory, 155 (2015) 124--134).
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davenport constant of the multiplicative semigroup of the Quotient Ring frac f_p x langle f x rangle
arXiv: Number Theory, 2014Co-Authors: Haoli Wang, Lizhen Zhang, Qinghong WangAbstract:Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial Ring in one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the Quotient Ring $\frac{\F_p[x]}{\langle f(x)\rangle}$. Among other results, we mainly prove that, for any prime $p>2$ and any polynomial $f(x)\in \F_p[x]$ which can be factorized into several pairwise non-associted irreducible polynomials in $\F_p[x]$, then $$D(\mathcal{S}_{f(x)}^p)=D(U(\mathcal{S}_{f(x)}^p)),$$ where $\mathcal{S}_{f(x)}^p$ denotes the multiplicative semigroup of the Quotient Ring $\frac{\F_p[x]}{\langle f(x)\rangle}$ and $U(\mathcal{S}_{f(x)}^p)$ denotes the group of units of the semigroup $\mathcal{S}_{f(x)}^p$.
Qinghong Wang - One of the best experts on this subject based on the ideXlab platform.
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note on the davenport constant of the multiplicative semigroup of the Quotient Ring x1d53d p x f x
International Journal of Number Theory, 2016Co-Authors: Haoli Wang, Lizhen Zhang, Qinghong WangAbstract:Let 𝒮 be a finite commutative semigroup. The Davenport constant of 𝒮, denoted D(𝒮), is defined to be the least positive integer l such that every sequence T of elements in 𝒮 of length at least l contains a proper subsequence T′ with the sum of all terms from T′ equaling the sum of all terms from T. Let 𝔽p[x] be a polynomial Ring in one variable over the prime field 𝔽p, and let f(x) ∈ 𝔽p[x]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the Quotient Ring 𝔽p[x] 〈f(x)〉 and proved that, for any prime p > 2 and any polynomial f(x) ∈ 𝔽p[x] which factors into a product of pairwise non-associate irreducible polynomials, D(𝒮f(x)p) = D(U(𝒮 f(x)p)), where 𝒮f(x)p denotes the multiplicative semigroup of the Quotient Ring 𝔽p[x] 〈f(x)〉 and U(𝒮f(x)p) denotes the group of units of the semigroup 𝒮f(x)p.
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davenport constant of the multiplicative semigroup of the Quotient Ring frac f_p x langle f x rangle
arXiv: Number Theory, 2014Co-Authors: Haoli Wang, Lizhen Zhang, Qinghong WangAbstract:Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial Ring in one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the Quotient Ring $\frac{\F_p[x]}{\langle f(x)\rangle}$. Among other results, we mainly prove that, for any prime $p>2$ and any polynomial $f(x)\in \F_p[x]$ which can be factorized into several pairwise non-associted irreducible polynomials in $\F_p[x]$, then $$D(\mathcal{S}_{f(x)}^p)=D(U(\mathcal{S}_{f(x)}^p)),$$ where $\mathcal{S}_{f(x)}^p$ denotes the multiplicative semigroup of the Quotient Ring $\frac{\F_p[x]}{\langle f(x)\rangle}$ and $U(\mathcal{S}_{f(x)}^p)$ denotes the group of units of the semigroup $\mathcal{S}_{f(x)}^p$.
