Let $K$ be a field and $S=K[x_1,ldots,x_n]$ a standard polynomial ring over $K$. In this paper, we give new combinatorial algorithms to compute the smallest $t$-spread lexicographic set and the smallest $t$-spread strongly stable set containing a given set of $t$-spread monomials of $S$. Some technical tools allowing to compute the cardinality of $t$-spread strongly stable sets avoiding their construction are also presented. Such functions are also implemented in a emph{Macaulay2} package, texttt{TSpreadIdeals}, to ease the computation of well-known results about algebraic invariants for $t$-spread ideals.
{"title":"Computational methods for $t$-spread monomial ideals","authors":"Luca Amata","doi":"10.24330/ieja.1402973","DOIUrl":"https://doi.org/10.24330/ieja.1402973","url":null,"abstract":"Let $K$ be a field and $S=K[x_1,ldots,x_n]$ a standard polynomial ring over $K$. In this paper, we give new combinatorial algorithms to compute the smallest $t$-spread lexicographic set and the smallest $t$-spread strongly stable set containing a given set of $t$-spread monomials of $S$. Some technical tools allowing to compute the cardinality of $t$-spread strongly stable sets avoiding their construction are also presented. Such functions are also implemented in a emph{Macaulay2} package, texttt{TSpreadIdeals}, to ease the computation of well-known results about algebraic invariants for $t$-spread ideals.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"24 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139269650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $K$ be a field and $K[x_1,x_{2}]$ the polynomial ring in two variables over $K$ with each $x_i$ of degree $1$. Let $L$ be the generalized mixed product ideal induced by a monomial ideal $Isubset K[x_1,x_2]$, where the ideals substituting the monomials in $I$ are squarefree Veronese ideals. In this paper, we study the integral closure of $L$, and the normality of $mathcal{R}(L)$, the Rees algebra of $L$. Furthermore, we give a geometric description of the integral closure of $mathcal{R}(L)$.
{"title":"Normality of Rees algebras of generalized mixed product ideals","authors":"M. La Barbiera, R. Moghimipor","doi":"10.24330/ieja.1402961","DOIUrl":"https://doi.org/10.24330/ieja.1402961","url":null,"abstract":"Let $K$ be a field and $K[x_1,x_{2}]$ the polynomial ring in two variables over $K$ with each $x_i$ of degree $1$. Let $L$ be the generalized mixed product ideal induced by a monomial ideal $Isubset K[x_1,x_2]$, where the ideals substituting the monomials in $I$ are squarefree Veronese ideals. In this paper, we study the integral closure of $L$, and the normality of $mathcal{R}(L)$, the Rees algebra of $L$. Furthermore, we give a geometric description of the integral closure of $mathcal{R}(L)$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"54 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139311249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $R$ be a ring and $n$ a fixed positive integer. A right $R$-module $M$ is called strongly $J$-$n$-injective if every $R$-homomorphism from an $n$-generated small submodule of a free right $R$-module $F$ to $M$ extends to a homomorphism of $F$ to $M$; a right $R$-module $V$ is said to be strongly $J$-$n$-flat, if for every $n$-generated small submodule $T$ of a free left $R$-module $F$, the canonical map $Votimes Trightarrow Votimes F$ is monic; a ring $R$ is called left strongly $J$-$n$-coherent if every $n$-generated small submodule of a free left $R$-module is finitely presented; a ring $R$ is said to be left $J$-$n$-semihereditary if every $n$-generated small left ideal of $R$ is projective. We study strongly $J$-$n$-injective modules, strongly $J$-$n$-flat modules and left strongly $J$-$n$-coherent rings. Using the concepts of strongly $J$-$n$-injectivity and strongly $J$-$n$-flatness of modules, we also present some characterizations of strongly $J$-$n$-coherent rings and $J$-$n$-semihereditary rings.
{"title":"Strongly J-n-Coherent rings","authors":"Zhanmin Zhu","doi":"10.24330/ieja.1411161","DOIUrl":"https://doi.org/10.24330/ieja.1411161","url":null,"abstract":"Let $R$ be a ring and $n$ a fixed positive integer. A right $R$-module $M$ is called strongly $J$-$n$-injective if every $R$-homomorphism from an $n$-generated small submodule of a free right $R$-module $F$ to $M$ extends to a homomorphism of $F$ to $M$; a right $R$-module $V$ is said to be strongly $J$-$n$-flat, if for every $n$-generated small submodule $T$ of a free left $R$-module $F$, the canonical map $Votimes Trightarrow Votimes F$ is monic; a ring $R$ is called left strongly $J$-$n$-coherent if every $n$-generated small submodule of a free left $R$-module is finitely presented; a ring $R$ is said to be left $J$-$n$-semihereditary if every $n$-generated small left ideal of $R$ is projective. We study strongly $J$-$n$-injective modules, strongly $J$-$n$-flat modules and left strongly $J$-$n$-coherent rings. Using the concepts of strongly $J$-$n$-injectivity and strongly $J$-$n$-flatness of modules, we also present some characterizations of strongly $J$-$n$-coherent rings and $J$-$n$-semihereditary rings.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"226 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139312309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $M=oplus_{nin mathbb{Z}}M_{n}$ be a strongly graded module over strongly graded ring $D=oplus_{nin mathbb{Z}} D_{n}$. In this paper, we prove that if $M_{0}$ is a unique factorization module (UFM for short) over $D_{0}$ and $D$ is a unique factorization domain (UFD for short), then $M$ is a UFM over $D$. Furthermore, if $D_{0}$ is a Noetherian domain, we give a necessary and sufficient condition for a positively graded module $L=oplus_{nin mathbb{Z}_{0}}M_{n}$ to be a UFM over positively graded domain $R=oplus_{nin mathbb{Z}_{0}}D_{n}$.
{"title":"Strongly Graded Modules and Positively Graded Modules which are Unique Factorization Modules","authors":"Iwan Ernanto, Indah E. Wijayanti, Akira Ueda","doi":"10.24330/ieja.1404435","DOIUrl":"https://doi.org/10.24330/ieja.1404435","url":null,"abstract":"Let $M=oplus_{nin mathbb{Z}}M_{n}$ be a strongly graded module over strongly graded ring $D=oplus_{nin mathbb{Z}} D_{n}$. In this paper, we prove that if $M_{0}$ is a unique factorization module (UFM for short) over $D_{0}$ and $D$ is a unique factorization domain (UFD for short), then $M$ is a UFM over $D$. Furthermore, if $D_{0}$ is a Noetherian domain, we give a necessary and sufficient condition for a positively graded module $L=oplus_{nin mathbb{Z}_{0}}M_{n}$ to be a UFM over positively graded domain $R=oplus_{nin mathbb{Z}_{0}}D_{n}$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"27 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139317380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. Mani̇kandan, Perumal Ramachandran, P. Madhusoodhanan
In this paper, we introduce the classes of $alpha$ and strictly-$alpha$ seminearrings and establishes some of their properties, mostly in relation to the possession of a mate function. Then we get the criterion for an $alpha$-seminearring to become a strictly-$alpha$ seminearring. We also obtain a complete characterisations of $alpha$ and strictly-$alpha$ seminearrings and proved certain results for $alpha$ and strictly-$alpha$ seminearrings via certain unique classes of seminearrings.
{"title":"The structure of certain unique classes of seminearrings","authors":"G. Mani̇kandan, Perumal Ramachandran, P. Madhusoodhanan","doi":"10.24330/ieja.1402798","DOIUrl":"https://doi.org/10.24330/ieja.1402798","url":null,"abstract":"In this paper, we introduce the classes of $alpha$ and strictly-$alpha$ seminearrings and establishes some of their properties, mostly in relation to the possession of a mate function. Then we get the criterion for an $alpha$-seminearring to become a strictly-$alpha$ seminearring. We also obtain a complete characterisations of $alpha$ and strictly-$alpha$ seminearrings and proved certain results for $alpha$ and strictly-$alpha$ seminearrings via certain unique classes of seminearrings.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"17 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139317629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One of the important classes of modules is the class of multiplication modules over a commutative ring. This topic has been considered by many authors and numerous results have been obtained in this area. After that, Tuganbaev also considered the multiplication module over a noncommutative ring. In this paper, we continue to consider the automorphism-invariance of multiplication modules over a noncommutative ring. We prove that if $R$ is a right duo ring and $M$ is a multiplication, finitely generated right $R$-module with a generating set ${m_1, dots , m_n}$ such that $r(m_i) = 0$ and $[m_iR: M] subseteq C(R)$ the center of $R$, then $M$ is projective. Moreover, if $R$ is a right duo, left quasi-duo, CMI ring and $M$ is a multiplication, non-singular, automorphism-invariant, finitely generated right $R$-module with a generating set ${m_1, dots , m_n}$ such that $r(m_i) = 0$ and $[m_iR: M] subseteq C(R)$ the center of $R$, then $M_R cong R$ is injective.
{"title":"On Automorphism-invariant multiplication modules over a noncommutative ring","authors":"L. Thuyet, T. C. Quynh","doi":"10.24330/ieja.1411145","DOIUrl":"https://doi.org/10.24330/ieja.1411145","url":null,"abstract":"One of the important classes of modules is the class of multiplication modules over a commutative ring. This topic has been considered by many authors and numerous results have been obtained in this area. After that, Tuganbaev also considered the multiplication module over a noncommutative ring. In this paper, we continue to consider the automorphism-invariance of multiplication modules over a noncommutative ring. We prove that if $R$ is a right duo ring and $M$ is a multiplication, finitely generated right $R$-module with a generating set ${m_1, dots , m_n}$ such that $r(m_i) = 0$ and $[m_iR: M] subseteq C(R)$ the center of $R$, then $M$ is projective. Moreover, if $R$ is a right duo, left quasi-duo, CMI ring and $M$ is a multiplication, non-singular, automorphism-invariant, finitely generated right $R$-module with a generating set ${m_1, dots , m_n}$ such that $r(m_i) = 0$ and $[m_iR: M] subseteq C(R)$ the center of $R$, then $M_R cong R$ is injective.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"13 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139323334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated Noetherian dimension of an $R$-module, where $R$ is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension and Noetherian dimension.
They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules.
It is proved that a quotient finite dimensional module $M$ has uncountably generated Krull dimension if and only if it has Krull dimension, but
the values of these dimensions might differ.
Similarly, a quotient finite dimensional module $M$ has uncountably generated Noetherian dimension if and only if it has Noetherian dimension.
We also show that the Noetherian dimension of a quotient finite dimensional module $M$ with uncountably generated Noetherian dimension $beta$ is less than or equal to $omega _{1}+beta $, where $omega_{1}$ is the first uncountable ordinal number.
{"title":"Dimension of uncountably generated submodules","authors":"Maryam DAVOUDİAN","doi":"10.24330/ieja.1385180","DOIUrl":"https://doi.org/10.24330/ieja.1385180","url":null,"abstract":"In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated Noetherian dimension of an $R$-module, where $R$ is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension and Noetherian dimension.
 They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules.
 It is proved that a quotient finite dimensional module $M$ has uncountably generated Krull dimension if and only if it has Krull dimension, but
 the values of these dimensions might differ.
 Similarly, a quotient finite dimensional module $M$ has uncountably generated Noetherian dimension if and only if it has Noetherian dimension.
 We also show that the Noetherian dimension of a quotient finite dimensional module $M$ with uncountably generated Noetherian dimension $beta$ is less than or equal to $omega _{1}+beta $, where $omega_{1}$ is the first uncountable ordinal number.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135875935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. E. Atani, M. Khoramdel, Saboura DOLATİ PİSHHESARİ
In this paper, we introduce the notion of pseudo-absorbing comultiplication modules. A full description of all indecomposable pseudo-absorbing comultiplication modules with finite dimensional top over certain kinds of pullback rings are given and establish a connection between the pseudo-absorbing comultiplication modules and the pure-injective modules over such rings.
{"title":"Pseudo-absorbing comultiplication modules over a pullback ring","authors":"S. E. Atani, M. Khoramdel, Saboura DOLATİ PİSHHESARİ","doi":"10.24330/ieja.1404416","DOIUrl":"https://doi.org/10.24330/ieja.1404416","url":null,"abstract":"In this paper, we introduce the notion of pseudo-absorbing comultiplication modules. A full description of all indecomposable pseudo-absorbing comultiplication modules with finite dimensional top over certain kinds of pullback rings are given and establish a connection between the pseudo-absorbing comultiplication modules and the pure-injective modules over such rings.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"25 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139354334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we define algebraic Lie algebra bundles, discuss some results on algebraic Lie algebra bundles and derivations of Lie algebra bundles. Some results involving inner derivations and central derivations of Lie algebra bundles are obtained.
{"title":"Algebraic Lie Algebra Bundles and Derivations of Lie Algebra Bundles","authors":"M. V. Moni̇ca, R. Rajendra","doi":"10.24330/ieja.1377714","DOIUrl":"https://doi.org/10.24330/ieja.1377714","url":null,"abstract":"In this paper, we define algebraic Lie algebra bundles, discuss some results on algebraic Lie algebra bundles and derivations of Lie algebra bundles. Some results involving inner derivations and central derivations of Lie algebra bundles are obtained.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"41 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139359567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the BGG category $O$ of a quantized universal enveloping algebra $U_q(mathfrak{g})$. It is well-known that $Motimes Nin O$ if $M$ or $N$ is finite dimensional. When $mathfrak{g}$ is simple and of type ADE, we prove in this paper that $Motimes Nnotin O$ if $M$ and $N$ are both infinite dimensional.
{"title":"Tensor products of infinite dimensional modules in the BGG category of a quantized simple Lie algebra of type ADE","authors":"Zhaoting Wei̇","doi":"10.24330/ieja.1357059","DOIUrl":"https://doi.org/10.24330/ieja.1357059","url":null,"abstract":"We consider the BGG category $O$ of a quantized universal enveloping algebra $U_q(mathfrak{g})$. It is well-known that $Motimes Nin O$ if $M$ or $N$ is finite dimensional. When $mathfrak{g}$ is simple and of type ADE, we prove in this paper that $Motimes Nnotin O$ if $M$ and $N$ are both infinite dimensional.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45302004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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