We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R...We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.展开更多
Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how ...Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how to construct left cotorsion pais from left n-cotorsion pairs,and establish a one-to-one correspondence between left Frobenius pairs and left(weak)Auslander-Buchweitz contexts.Some applications are given in the Gorenstein homological theory of triangulated categories.展开更多
Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in parti...Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in particular,R is a symmetric ring if and only if R is a(1,1)-symmetric ring.We show that e∈E(R)is left semicentral if and only if R is a(1−e,e)-symmetric ring;in particular,R is an Abel ring if and only if R is a(1−e,e)-symmetric ring for each e∈E(R).We also show that R is(g,e)-symmetric if and only if ge∈E(R),geRge is symmetric,and gxye=gxeye=gxgye for any x,y∈R.Using e-symmetric rings,we construct some new classes of rings.展开更多
Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions...Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions.Finally,we give some examples to illustrate that both the upper and lower bounds can be reached.展开更多
The notion of D-mutation pairs of subcategories in an n-exangulated category is defined in this article.When(Z,Z)is a P-mutation pair in an n-exangulated category(C,E,s),the quotient category Z/Dcarries naturally an(n...The notion of D-mutation pairs of subcategories in an n-exangulated category is defined in this article.When(Z,Z)is a P-mutation pair in an n-exangulated category(C,E,s),the quotient category Z/Dcarries naturally an(n+2)-angulated structure.This result generalizes a theorem of Zhou and Zhu for extriangulated categories.展开更多
We investigate how the category of comodules of bimonads can be made into a monoidal category.It suffices that the monad and comonad in question are bimonads,with some extra compatibility relation.On a monoidal catego...We investigate how the category of comodules of bimonads can be made into a monoidal category.It suffices that the monad and comonad in question are bimonads,with some extra compatibility relation.On a monoidal category of comodules of bimonads,we cons true t a braiding and get the necessary and sufficien t conditions making it a braided monoidal category.As an application,we consider the category of comodules of corings and the category of entwined modules.展开更多
The International Conference on Algebra and Related Topics(ICART2018),which was held at the Faculty of Sciences of the Mohammed V University in Rabat,Morocco,July 2-5,2018,welcomed more than 180 participants from diff...The International Conference on Algebra and Related Topics(ICART2018),which was held at the Faculty of Sciences of the Mohammed V University in Rabat,Morocco,July 2-5,2018,welcomed more than 180 participants from different nationalities.The participants presented lectures about active and interesting areas of research in algebra and related to pics.展开更多
Let R be a prime ring of characteristic different from 2 and 3,Qr be its right Martindale quotient ring and C be its extended centroid.Suppose that F and G are generalized skew derivations of R,L a non-central Lie ide...Let R be a prime ring of characteristic different from 2 and 3,Qr be its right Martindale quotient ring and C be its extended centroid.Suppose that F and G are generalized skew derivations of R,L a non-central Lie ideal of R and n≥1 a fixed positive integer.Under appropriate conditions we prove that if(F(x)x;—xG(x))^n=0 for all x∈L,then one of the following holds:(a)there exists c∈Qr such that F(x)=xc and G(e)=cx;(b)R satisfies S4 and there exist a,b,c∈Qr such that F(x)=ax+xc,G(x)=cx+xb and(a—b)^2=0.展开更多
The special issues ＂Ring Theory and Related Topics （I, II）＂ of Frontiers of Mathematics in China （FMC） are an outcome of the Seventh China-Japan- Korea International Conference on Ring Theory held from July 1 to...The special issues ＂Ring Theory and Related Topics （I, II）＂ of Frontiers of Mathematics in China （FMC） are an outcome of the Seventh China-Japan- Korea International Conference on Ring Theory held from July 1 to July 7, 2015 in Hangzhou, China.展开更多
文摘We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.
基金supported by the NSF of China(12001168,11901341,11971225,12171207)Henan University of Engineering(DKJ2019010)+1 种基金the Key Research Project of Education Department of Henan Province(21A110006)Youth Innovation Team of Universities of Shandong Province(2022KJ314).
文摘Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how to construct left cotorsion pais from left n-cotorsion pairs,and establish a one-to-one correspondence between left Frobenius pairs and left(weak)Auslander-Buchweitz contexts.Some applications are given in the Gorenstein homological theory of triangulated categories.
基金supported by the Foundation of Natural Science of China(12301029,11171291)Natural Science Fund for Colleges and Universities in Jiangsu Province(11KJB110019 and 15KJB110023).
文摘Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in particular,R is a symmetric ring if and only if R is a(1,1)-symmetric ring.We show that e∈E(R)is left semicentral if and only if R is a(1−e,e)-symmetric ring;in particular,R is an Abel ring if and only if R is a(1−e,e)-symmetric ring for each e∈E(R).We also show that R is(g,e)-symmetric if and only if ge∈E(R),geRge is symmetric,and gxye=gxeye=gxgye for any x,y∈R.Using e-symmetric rings,we construct some new classes of rings.
基金supported by the National Natural Science Foundation of China(11971255,11901567,12071120)supported by the Hunan Provincial Natural Science Foundation of China(2023JJ30008)the National Natural Science Foundation of China(12371034).
文摘Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions.Finally,we give some examples to illustrate that both the upper and lower bounds can be reached.
基金Supported by the National Natural Science Foundation of China(Grant No.11771212)a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘The notion of D-mutation pairs of subcategories in an n-exangulated category is defined in this article.When(Z,Z)is a P-mutation pair in an n-exangulated category(C,E,s),the quotient category Z/Dcarries naturally an(n+2)-angulated structure.This result generalizes a theorem of Zhou and Zhu for extriangulated categories.
基金Supported by the National Natural Science Foundation of China(No.11601486)Foundation of Zhejiang Educational Commitee(No.Y201738645)Project of Zhejiang College,Shanghai University of Finance and Economics(No.2018YJYB01).
文摘We investigate how the category of comodules of bimonads can be made into a monoidal category.It suffices that the monad and comonad in question are bimonads,with some extra compatibility relation.On a monoidal category of comodules of bimonads,we cons true t a braiding and get the necessary and sufficien t conditions making it a braided monoidal category.As an application,we consider the category of comodules of corings and the category of entwined modules.
文摘The International Conference on Algebra and Related Topics(ICART2018),which was held at the Faculty of Sciences of the Mohammed V University in Rabat,Morocco,July 2-5,2018,welcomed more than 180 participants from different nationalities.The participants presented lectures about active and interesting areas of research in algebra and related to pics.
文摘Let R be a prime ring of characteristic different from 2 and 3,Qr be its right Martindale quotient ring and C be its extended centroid.Suppose that F and G are generalized skew derivations of R,L a non-central Lie ideal of R and n≥1 a fixed positive integer.Under appropriate conditions we prove that if(F(x)x;—xG(x))^n=0 for all x∈L,then one of the following holds:(a)there exists c∈Qr such that F(x)=xc and G(e)=cx;(b)R satisfies S4 and there exist a,b,c∈Qr such that F(x)=ax+xc,G(x)=cx+xb and(a—b)^2=0.
文摘The special issues ＂Ring Theory and Related Topics （I, II）＂ of Frontiers of Mathematics in China （FMC） are an outcome of the Seventh China-Japan- Korea International Conference on Ring Theory held from July 1 to July 7, 2015 in Hangzhou, China.
