Pub Date : 2021-06-01DOI: 10.22124/JART.2021.18226.1241
F. Saraei
Let $M$ be a module over a commutative ring $R$, $Z_{*}(M)$ be its set of weak zero-divisor elements, andif $min M$, then let $I_m=(Rm:_R M)={rin R : rMsubseteq Rm}$. The annihilator graph of $M$ is the (undirected) graph$AG(M)$ with vertices $tilde{Z_{*}}(M)=Z_{*}(M)setminus {0}$, and two distinct vertices $m$ and $n$ are adjacent if andonly if $(0:_R I_{m}I_{n}M)neq (0:_R m)cup (0:_R n)$. We show that $AG(M)$ is connected with diameter at most two and girth at mostfour. Also, we study some properties of the zero-divisor graph of reduced multiplication-like $R$-modules.
设$M$是交换环$R$上的一个模，$Z_{*}(M)$是交换环$R$上的弱零因子元素集，如果$min M$，则设$I_m=(Rm:_R M)={rin R: rMsubseteq Rm}$。$M$的湮灭子图是顶点$ tide {Z_{*}}(M)=Z_{*}(M) set-{0}$的(无向)图$AG(M)$，且两个不同的顶点$M$和$n$相邻当且仅当$(0:_R I_{M}I_{n}M)neq (0:_R M) cup (0:_R n)$。我们证明了$AG(M)$与直径最多为2，周长最多为4。此外，我们还研究了类R模的约简乘法的零因子图的一些性质。
{"title":"The annihilator graph of modules over commutative rings","authors":"F. Saraei","doi":"10.22124/JART.2021.18226.1241","DOIUrl":"https://doi.org/10.22124/JART.2021.18226.1241","url":null,"abstract":"Let $M$ be a module over a commutative ring $R$, $Z_{*}(M)$ be its set of weak zero-divisor elements, andif $min M$, then let $I_m=(Rm:_R M)={rin R : rMsubseteq Rm}$. The annihilator graph of $M$ is the (undirected) graph$AG(M)$ with vertices $tilde{Z_{*}}(M)=Z_{*}(M)setminus {0}$, and two distinct vertices $m$ and $n$ are adjacent if andonly if $(0:_R I_{m}I_{n}M)neq (0:_R m)cup (0:_R n)$. We show that $AG(M)$ is connected with diameter at most two and girth at mostfour. Also, we study some properties of the zero-divisor graph of reduced multiplication-like $R$-modules.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"93-108"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49002892","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.16939.1210
A. Paad, A. Jafari
In this paper, the notions of $n$-fold obstinate and $n$-fold fantastic (pre)filterin $EQ$-algebras are introduced and the relationship among $n$-fold obstinate, maximal, $n$-fold fantastic, and $n$-fold (positive) implicative prefilters are investigated. Moreover, the quotient $EQ$-algebra induced by an $n$-fold obstinate filter is studied and it is proved that the quotient $EQ$-algebra induced byan $n$-fold fantastic filter of a good $EQ$-algebra with bottom element $0$ is an involutive $EQ$-algebra. Finally, the relationships between types of $n$-fold filters in residuated $EQ$-algebras is shown by diagrams
{"title":"$n$-fold obstinate and $n$-fold fantastic (pre)filters of $EQ$-algebras","authors":"A. Paad, A. Jafari","doi":"10.22124/JART.2021.16939.1210","DOIUrl":"https://doi.org/10.22124/JART.2021.16939.1210","url":null,"abstract":"In this paper, the notions of $n$-fold obstinate and $n$-fold fantastic (pre)filterin $EQ$-algebras are introduced and the relationship among $n$-fold obstinate, maximal, $n$-fold fantastic, and $n$-fold (positive) implicative prefilters are investigated. Moreover, the quotient $EQ$-algebra induced by an $n$-fold obstinate filter is studied and it is proved that the quotient $EQ$-algebra induced byan $n$-fold fantastic filter of a good $EQ$-algebra with bottom element $0$ is an involutive $EQ$-algebra. Finally, the relationships between types of $n$-fold filters in residuated $EQ$-algebras is shown by diagrams","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"31-50"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43902536","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.17227.1214
M. Adlifard, S. Payrovi
Let $R$ be a commutative ring and let $Bbb A(R)$ bethe set of all ideals of $R$ with nonzero annihilator.The annihilator-ideal graph of $R$ is defined as the graph${A_I}(R)$ with the vertex set $Bbb A(R)^*=Bbb A(R)setminus{0}$ and twodistinct vertices $I$ and $J$ are adjacent if and only if${rm Ann}_R(IJ)neq{rm Ann}_R(I) cup{rm Ann}_R(J)$. In this paper, perfectness of${A_I}(R)$ for some classes of rings is investigated.
{"title":"Some classes of perfect annihilator-ideal graphs associated with commutative rings","authors":"M. Adlifard, S. Payrovi","doi":"10.22124/JART.2021.17227.1214","DOIUrl":"https://doi.org/10.22124/JART.2021.17227.1214","url":null,"abstract":"Let $R$ be a commutative ring and let $Bbb A(R)$ bethe set of all ideals of $R$ with nonzero annihilator.The annihilator-ideal graph of $R$ is defined as the graph${A_I}(R)$ with the vertex set $Bbb A(R)^*=Bbb A(R)setminus{0}$ and twodistinct vertices $I$ and $J$ are adjacent if and only if${rm Ann}_R(IJ)neq{rm Ann}_R(I) cup{rm Ann}_R(J)$. In this paper, perfectness of${A_I}(R)$ for some classes of rings is investigated.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"21-29"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44110456","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.17609.1230
H. Hakmi
This paper is a continuation of study rings relative to rightideal, where we study the concepts of regular and local ringsrelative to right ideal. We give some relations between $P-$local($P-$regular) and local (regular) rings. New characterizationobtained include necessary and sufficient conditions of a ring $R$to be regular, local ring in terms $P-$regular, $P-$local ofmatrices ring $M_{2}(R)$. Also, We proved that every ring is localrelative to any maximal right ideal of it.
{"title":"P-Regular and P-Local Rings","authors":"H. Hakmi","doi":"10.22124/JART.2021.17609.1230","DOIUrl":"https://doi.org/10.22124/JART.2021.17609.1230","url":null,"abstract":"This paper is a continuation of study rings relative to rightideal, where we study the concepts of regular and local ringsrelative to right ideal. We give some relations between $P-$local($P-$regular) and local (regular) rings. New characterizationobtained include necessary and sufficient conditions of a ring $R$to be regular, local ring in terms $P-$regular, $P-$local ofmatrices ring $M_{2}(R)$. Also, We proved that every ring is localrelative to any maximal right ideal of it.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"1-19"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42060538","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.18893.1254
S. Rajaee
Let $R$ be a commutative ring with identity and let $M$ be a unitary $R$-module. In this paper, among various results, we prove that if $M$ is a cancellation $R$-module and $L$ is a nonzero simple submodule of $M$, then $L$ is a copure submodule of $M$. Moreover, in this case, if $M$ is co-m, then $M/L$ is also a co-m $R$-module. We investigate various conditions under which the quotient module $M/N$ of a co-m $M$ is also a co-m. We prove that if $M$ is a cancellation Noetherian co-m module, then for every second submodule $N$ of $M$ the quotient module $M/N$ is a co-m $R$-module. We obtain some results concerning socle and radical of co-m modules.
{"title":"Some results on the quotient of co-m modules","authors":"S. Rajaee","doi":"10.22124/JART.2021.18893.1254","DOIUrl":"https://doi.org/10.22124/JART.2021.18893.1254","url":null,"abstract":"Let $R$ be a commutative ring with identity and let $M$ be a unitary $R$-module. In this paper, among various results, we prove that if $M$ is a cancellation $R$-module and $L$ is a nonzero simple submodule of $M$, then $L$ is a copure submodule of $M$. Moreover, in this case, if $M$ is co-m, then $M/L$ is also a co-m $R$-module. We investigate various conditions under which the quotient module $M/N$ of a co-m $M$ is also a co-m. We prove that if $M$ is a cancellation Noetherian co-m module, then for every second submodule $N$ of $M$ the quotient module $M/N$ is a co-m $R$-module. We obtain some results concerning socle and radical of co-m modules.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"79-92"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45727675","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.18981.1256
F. Farzalipour, P. Ghiasvand
Let $R$ be a Krasner hyperring and $M$ be an $R$- hypermodule. Let $psi: S^{h}(M)rightarrow S^{h}(M)cup {emptyset}$ be a function, where $S^{h}(M)$ denote the set of all subhypermodules of $M$. In the first part of this paper, we introduce the concept of a secondary hypermodule over a Krasner hyperring. A non-zero hypermodule $M$ over a Krasner hyperring $R$ is called secondary if for every $rin R$, $rM=M$ or $r^{n}M=0$ for some positive integer $n$. Then we investigate some basic properties of secondary hypermodules. Second, we introduce the notion of $psi$-secondary subhypermodules of an $R$-hypermodule and we obtain some properties of such subhypermodules.
{"title":"On secondary subhypermodules","authors":"F. Farzalipour, P. Ghiasvand","doi":"10.22124/JART.2021.18981.1256","DOIUrl":"https://doi.org/10.22124/JART.2021.18981.1256","url":null,"abstract":"Let $R$ be a Krasner hyperring and $M$ be an $R$- hypermodule. Let $psi: S^{h}(M)rightarrow S^{h}(M)cup {emptyset}$ be a function, where $S^{h}(M)$ denote the set of all subhypermodules of $M$. In the first part of this paper, we introduce the concept of a secondary hypermodule over a Krasner hyperring. A non-zero hypermodule $M$ over a Krasner hyperring $R$ is called secondary if for every $rin R$, $rM=M$ or $r^{n}M=0$ for some positive integer $n$. Then we investigate some basic properties of secondary hypermodules. Second, we introduce the notion of $psi$-secondary subhypermodules of an $R$-hypermodule and we obtain some properties of such subhypermodules.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"143-158"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47229258","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.15730.1190
G. Cannon, V. Enlow
Let $(G, +)$ be a finite group, written additively with identity 0, but not necessarily abelian, and let $H$ be a nonzero, proper subgroup of $G$. Then the set $M = {f : G to G | f(G) subseteq H hbox{and} f(0) = 0 }$ is a right, zero-symmetric nearring under pointwise addition and function composition. We find necessary and sufficient conditions for $M$ to be a ring and determine all ideals of $M$, the center of $M$, and the distributive elements of $M$.
设$(G， +)$是一个有限群，以单位0相加，但不一定是阿贝尔群，且设$H$是$G$的非零真子群。则集合$M = {f: G to G | f(G) subseteq H hbox{and} f(0) = 0}$是一个点加法和函数复合下的右零对称近环。我们找到了$M$是环的充分必要条件，并确定了$M$的所有理想、$M$的中心和$M$的分配元。
{"title":"Nearrings of functions without identity determined by a single subgroup","authors":"G. Cannon, V. Enlow","doi":"10.22124/JART.2021.15730.1190","DOIUrl":"https://doi.org/10.22124/JART.2021.15730.1190","url":null,"abstract":"Let $(G, +)$ be a finite group, written additively with identity 0, but not necessarily abelian, and let $H$ be a nonzero, proper subgroup of $G$. Then the set $M = {f : G to G | f(G) subseteq H hbox{and} f(0) = 0 }$ is a right, zero-symmetric nearring under pointwise addition and function composition. We find necessary and sufficient conditions for $M$ to be a ring and determine all ideals of $M$, the center of $M$, and the distributive elements of $M$.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"121-129"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43389813","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.14595.1167
S. Billawria, Shallu Sharma
In this paper, we introduce a generalized form of the class of topological rings, namely -topological rings by using -open sets which itself is a generalized form of open sets. Translation of open(closed) sets and multiplication by invertible elements of open(closed) sets of the -topological rings are investigated. Some other useful results on -topological rings are also given. Examples of -topological rings which fails to be topological rings are also provided. We further de ne -topological rings with unity in the sequel and presented some results on it.
{"title":"On beta-topological rings","authors":"S. Billawria, Shallu Sharma","doi":"10.22124/JART.2021.14595.1167","DOIUrl":"https://doi.org/10.22124/JART.2021.14595.1167","url":null,"abstract":"In this paper, we introduce a generalized form of the class of topological rings, namely -topological rings by using -open sets which itself is a generalized form of open sets. Translation of open(closed) sets and multiplication by invertible elements of open(closed) sets of the -topological rings are investigated. Some other useful results on -topological rings are also given. Examples of -topological rings which fails to be topological rings are also provided. We further de ne -topological rings with unity in the sequel and presented some results on it.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"51-60"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42407302","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.17550.1226
J. J. Patil, K. Pawar
The $Gamma$-semihyperring is a generalization of the concepts of a semiring, a semihyperring and a $Gamma$-semiring. In this paper, the notions of completely prime ideals and prime radicals for $Gamma$-semihyperring are introduced and studied some important properties accordingly. We also introduced the notions of $m$-system and complete $m$-system. Then characterizations of prime ideals and completely prime ideals of $Gamma$-semihyperring with the help of $m$-system and complete $m$-system has been taken into account. It is our attempt to find a bridge between semiprime (completely semiprime) ideals and prime (completely prime) ideals of a $Gamma$-semihyperring.
{"title":"On Prime and semiprime ideals of $Gamma$-semihyperrings","authors":"J. J. Patil, K. Pawar","doi":"10.22124/JART.2021.17550.1226","DOIUrl":"https://doi.org/10.22124/JART.2021.17550.1226","url":null,"abstract":"The $Gamma$-semihyperring is a generalization of the concepts of a semiring, a semihyperring and a $Gamma$-semiring. In this paper, the notions of completely prime ideals and prime radicals for $Gamma$-semihyperring are introduced and studied some important properties accordingly. We also introduced the notions of $m$-system and complete $m$-system. Then characterizations of prime ideals and completely prime ideals of $Gamma$-semihyperring with the help of $m$-system and complete $m$-system has been taken into account. It is our attempt to find a bridge between semiprime (completely semiprime) ideals and prime (completely prime) ideals of a $Gamma$-semihyperring.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"131-141"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43901066","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}
Pub Date : 2021-06-01DOI: 10.22124/JART.2021.18801.1252
A. Estaji, M. R. Sarpoushi
Let $mathcal{R}_c( L)$ be the pointfree version of $C_c(X)$, the subring of $C(X)$ whose elements have countable image. We shall call a frame $L $ a $CP$-frame if thering $mathcal{R}_c( L)$ is regular. % The main aim of this paper is to introduce $CP$-frames, that is $mathcal{R}_c( L)$ is a regular ring. We give some We give some characterizations of $CP$-frames and we show that $L$ is a $CP$-frame if and only if each prime ideal of $mathcal{R}_c ( L)$ is an intersection of maximal ideals if and only if every ideal of $mathcal{R}_c ( L)$ is a $z_c$-ideal. In particular, we prove that any $P$-frame is a $CP$-frame but not conversely, in general. In addition, we study some results about $CP$-frames like the relation between a $CP$-frame $L$ and ideals of closed quotients of $L$. Next, we characterize $CP$-frames as precisely those $L$ for which every prime ideal in the ring $mathcal{R}_c ( L)$ is a $z_c$-ideal. Finally, we show that this characterization still holds if prime ideals are replaced by essential ideals, radical ideals, convex ideals, or absolutely convex ideals.
{"title":"On CP-frames","authors":"A. Estaji, M. R. Sarpoushi","doi":"10.22124/JART.2021.18801.1252","DOIUrl":"https://doi.org/10.22124/JART.2021.18801.1252","url":null,"abstract":"Let $mathcal{R}_c( L)$ be the pointfree version of $C_c(X)$, the subring of $C(X)$ whose elements have countable image. We shall call a frame $L $ a $CP$-frame if thering $mathcal{R}_c( L)$ is regular. % The main aim of this paper is to introduce $CP$-frames, that is $mathcal{R}_c( L)$ is a regular ring. We give some We give some characterizations of $CP$-frames and we show that $L$ is a $CP$-frame if and only if each prime ideal of $mathcal{R}_c ( L)$ is an intersection of maximal ideals if and only if every ideal of $mathcal{R}_c ( L)$ is a $z_c$-ideal. In particular, we prove that any $P$-frame is a $CP$-frame but not conversely, in general. In addition, we study some results about $CP$-frames like the relation between a $CP$-frame $L$ and ideals of closed quotients of $L$. Next, we characterize $CP$-frames as precisely those $L$ for which every prime ideal in the ring $mathcal{R}_c ( L)$ is a $z_c$-ideal. Finally, we show that this characterization still holds if prime ideals are replaced by essential ideals, radical ideals, convex ideals, or absolutely convex ideals.","PeriodicalId":52302,"journal":{"name":"Journal of Algebra and Related Topics","volume":"9 1","pages":"109-119"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43169262","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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