Let $F_0$ be an absolutely algebraic field of characteristic $p>0$ and $kappa$ an infinite cardinal. It is shown that there exists a field $F$ such that $F^*cong F^*_0oplus(oplus_kappa mathbb{Q})$ with $Br(F)={0}$. Let $L$ be an algebraic closure of $F$. Then for any finite subextension $K$ of $L/F$, we have $K^*cong T(K^*)oplus(oplus_kappa mathbb{Q})$, where $T(K^*)$ is the group of torsion elements of $K^*$. In addition, $Br(K)={0}$ and $[K:F]=[T(K^*) cup {0}:F_0]$.
设$F_0$为特征为$p>0$的绝对代数域，$kappa$为无限基数。结果表明，存在一个域$F$，使得$F^*cong F^*_0oplus(oplus_kappa mathbb{Q})$与$Br(F)={0}$。设$L$为$F$的代数闭包。然后对于$L/F$的任意有限子扩展$K$，我们有$K^*cong T(K^*)oplus(oplus_kappa mathbb{Q})$，其中$T(K^*)$是$K^*$的扭转单元群。此外，还有$Br(K)={0}$和$[K:F]=[T(K^*) cup {0}:F_0]$。
{"title":"Fields whose torsion free parts divisible with trivial Brauer group","authors":"R. Fallah-Moghaddam","doi":"10.24330/ieja.1144156","DOIUrl":"https://doi.org/10.24330/ieja.1144156","url":null,"abstract":"Let $F_0$ be an absolutely algebraic field of characteristic $p>0$ and \u0000$kappa$ an infinite cardinal. It is shown that there exists a \u0000field $F$ such that $F^*cong F^*_0oplus(oplus_kappa \u0000mathbb{Q})$ with $Br(F)={0}$. Let $L$ be an algebraic closure \u0000of $F$. Then for any finite subextension $K$ of $L/F$, we have \u0000$K^*cong T(K^*)oplus(oplus_kappa mathbb{Q})$, where $T(K^*)$ \u0000is the group of torsion elements of $K^*$. In addition, \u0000$Br(K)={0}$ and $[K:F]=[T(K^*) cup {0}:F_0]$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43131691","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 prime ring, $Q_r$ its right Martindale quotient ring, $L$ a non-central Lie ideal of $R$, $ngeq 1$ a fixed integer, $F$ and $G$ two generalized skew derivations of $R$ with the same associated automorphism, $pin R$ a fixed element. If $pbigl(F(x)F(y)-G(y)xbigr)^n=0$, for any $x,y in L$, then there exist $a,cin Q_r$ such that $F(x)=ax$ and $G(x)=cx$, for any $xin R$, with $pa=pc=0$, unless when $R$ satisfies the standard polynomial identity $s_4(x_1,ldots,x_4)$.
{"title":"Annihilator conditions with generalized skew derivations and Lie ideals of prime rings","authors":"V. De Filippis, N. Rehman, G. Scudo","doi":"10.24330/ieja.1143810","DOIUrl":"https://doi.org/10.24330/ieja.1143810","url":null,"abstract":"Let $R$ be a prime ring, $Q_r$ its right Martindale quotient ring, $L$ a non-central Lie ideal of $R$, $ngeq 1$ a fixed integer, $F$ and $G$ two generalized skew derivations of $R$ with the same associated automorphism, $pin R$ a fixed element. If $pbigl(F(x)F(y)-G(y)xbigr)^n=0$, for any $x,y in L$, then there exist $a,cin Q_r$ such that $F(x)=ax$ and $G(x)=cx$, for any $xin R$, with $pa=pc=0$, unless when $R$ satisfies the standard polynomial identity $s_4(x_1,ldots,x_4)$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41729569","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 present many new characterizations of strongly EP elements in rings with involution. We especially investigate the strongly EP elements by constructing certain equations and considering the solutions of equations, revealing the existence of solutions of certain equations and the general solutions of some binary equations that play a role in characterizing strongly EP elements. Proofs of relevant conclusions are also given.
{"title":"Strongly EP elements and the solutions of equations in rings","authors":"Yue Sui, Yimin Huang, Junchao Wei","doi":"10.24330/ieja.1143740","DOIUrl":"https://doi.org/10.24330/ieja.1143740","url":null,"abstract":"In this paper, we present many new characterizations of strongly EP elements in rings with involution. We especially investigate the strongly EP elements by constructing certain equations and considering the solutions of equations, revealing the existence of solutions of certain equations and the general solutions of some binary equations that play a role in characterizing strongly EP elements. Proofs of relevant conclusions are also given.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46356851","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 study symmetric algebras $A$ over a field $F$ in which the Jacobson radical of the center of $A$, the socle of the center of $A$ or the Reynolds ideal of $A$ are ideals.
{"title":"Ideals in the center of symmetric algebras","authors":"Sofia Brenner, B. Kulshammer","doi":"10.24330/ieja.1295669","DOIUrl":"https://doi.org/10.24330/ieja.1295669","url":null,"abstract":"We study symmetric algebras $A$ over a field $F$ in which the Jacobson radical of the center of $A$, the socle of the center of $A$ or the Reynolds ideal of $A$ are ideals.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44067087","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}
The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and Hom-alternative algebras, their structure is defined with two commuting multiplicative linear maps. We study cohomology and one-parameter formal deformation theory of left BiHom-alternative algebras. Moreover, we study central and $T_theta$-extensions of BiHom-alternative algebras and their relationship with cohomology. Finally, we investigate generalized derivations and give some relevant results.
{"title":"Deformations and Extensions of BiHom-alternative algebras","authors":"T. Chtioui, S. Mabrouk, A. Makhlouf","doi":"10.24330/ieja.1260492","DOIUrl":"https://doi.org/10.24330/ieja.1260492","url":null,"abstract":"The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and \u0000Hom-alternative algebras, their structure is defined with two \u0000commuting multiplicative linear maps. We study cohomology and \u0000one-parameter formal deformation theory of left BiHom-alternative \u0000algebras. Moreover, we study central and $T_theta$-extensions of \u0000BiHom-alternative algebras and their relationship with \u0000cohomology. \u0000Finally, we investigate generalized derivations and give some relevant results.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48689183","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}
This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.
{"title":"Applications of reduced and coreduced modules I","authors":"D. Ssevviiri","doi":"10.24330/ieja.1299587","DOIUrl":"https://doi.org/10.24330/ieja.1299587","url":null,"abstract":"This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48204020","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}
. An ideal I of a commutative ring is called a cancellation ideal if IB = IC implies B = C for all ideals B and C . Let D be a principal ideal domain (PID), a,b ∈ D be nonzero elements with a (cid:45) b , ( a,b ) D = dD for some d ∈ D , D a = D/aD be the quotient ring of D modulo aD , and bD a = ( a,b ) D/aD ; so bD a is a nonzero commutative ring. In this paper, we show that the following three properties are equivalent: (i) ad is a prime element and a (cid:45) d 2 , (ii) every nonzero ideal of bD a is a cancellation ideal, and (iii) bD a is a field.
。如果IB = IC对所有理想B和C都意味着B = C，则交换环的理想I称为抵消理想。设D是主理想域(PID)， a,b∈D是非零元素，且a (cid:45) b， (a,b) D = dD，对于某些D∈D, da = D/aD是D模aD的商环，且bD a = (a,b) D/aD;所以bda是一个非零交换环。本文证明了下列三个性质是等价的:(i) ad是素元，a (cid:45) d2， (ii) bda的每一个非零理想都是抵消理想，(iii) bda是一个域。
{"title":"When Does a Quotient Ring of a PID Have the Cancellation Property?","authors":"G. Chang, J. Oh","doi":"10.24330/ieja.1102363","DOIUrl":"https://doi.org/10.24330/ieja.1102363","url":null,"abstract":". An ideal I of a commutative ring is called a cancellation ideal if IB = IC implies B = C for all ideals B and C . Let D be a principal ideal domain (PID), a,b ∈ D be nonzero elements with a (cid:45) b , ( a,b ) D = dD for some d ∈ D , D a = D/aD be the quotient ring of D modulo aD , and bD a = ( a,b ) D/aD ; so bD a is a nonzero commutative ring. In this paper, we show that the following three properties are equivalent: (i) ad is a prime element and a (cid:45) d 2 , (ii) every nonzero ideal of bD a is a cancellation ideal, and (iii) bD a is a field.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45563825","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}
. The main objective of this paper is to study (quasi-)morphic property of skew polynomial rings. Let R be a ring, σ be a ring homomorphism on R and n ≥ 1. We show that R inherits the quasi-morphic property from R [ x ; σ ] / ( x n +1 ). It is also proved that the morphic property over R [ x ; σ ] / ( x n +1 ) implies that R is a regular ring. Moreover, we characterize a unit-regular ring R via the morphic property of R [ x ; σ ] / ( x n +1 ). We also investigate the relationship between strongly regular rings and centrally morphic rings. For instance, we show that for a domain R , R [ x ; σ ] / ( x n +1 ) is (left) centrally morphic if and only if R is a division ring and σ ( r ) = u − 1 ru for some u ∈ R . Examples which delimit and illustrate our results are provided.
． 本文的主要目的是研究歪斜多项式环的(拟)态性质。设R为环，σ为R上n≥1的环同态。我们证明了R继承了R [x]的拟态性质;σ] / (x n +1)并证明了R [x]上的态性;σ] / (x n +1)意味着R是正则环。此外，我们利用R [x]的态性质刻画了一个单位正则环R;σ] / (x n +1)我们还研究了强正则环和中心态环之间的关系。例如，对于定义域R, R [x;σ] / (x n +1)是(左)中心态的当且仅当R是一个除环并且σ (R) = u−1 ru对于某个u∈R。给出了界定和说明我们的结果的例子。
{"title":"On (quasi-)morphic property of skew polynomial rings","authors":"N. Dehghani","doi":"10.24330/ieja.1102387","DOIUrl":"https://doi.org/10.24330/ieja.1102387","url":null,"abstract":". The main objective of this paper is to study (quasi-)morphic property of skew polynomial rings. Let R be a ring, σ be a ring homomorphism on R and n ≥ 1. We show that R inherits the quasi-morphic property from R [ x ; σ ] / ( x n +1 ). It is also proved that the morphic property over R [ x ; σ ] / ( x n +1 ) implies that R is a regular ring. Moreover, we characterize a unit-regular ring R via the morphic property of R [ x ; σ ] / ( x n +1 ). We also investigate the relationship between strongly regular rings and centrally morphic rings. For instance, we show that for a domain R , R [ x ; σ ] / ( x n +1 ) is (left) centrally morphic if and only if R is a division ring and σ ( r ) = u − 1 ru for some u ∈ R . Examples which delimit and illustrate our results are provided.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48213498","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}
{"title":"The effect of $S$-accr on intermediate rings between certain pairs of rings","authors":"S. Visweswaran","doi":"10.24330/ieja.1096895","DOIUrl":"https://doi.org/10.24330/ieja.1096895","url":null,"abstract":"","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41638157","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}
{"title":"Graded S-1-absorbing prime submodules in graded multiplication modules","authors":"F. Farzalipour, P. Ghiasvand","doi":"10.24330/ieja.1081701","DOIUrl":"https://doi.org/10.24330/ieja.1081701","url":null,"abstract":"","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46034183","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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