Let R be a ring with an endomorphismσ,F∪{0}the free monoid generated by U={u1,…,ut}with 0 added,and M a factor of F obtained by setting certain monomials in F to 0 such that M^(n)=0 for some n.Then we can form the ...Let R be a ring with an endomorphismσ,F∪{0}the free monoid generated by U={u1,…,ut}with 0 added,and M a factor of F obtained by setting certain monomials in F to 0 such that M^(n)=0 for some n.Then we can form the non-semiprime skew monoid ring R[M;σ].A local ring R is called bleached if for any j∈J(R)and any u∈U(R),the abelian group endomorphisms l_(u)−r_(j) and l_(j)−r_(u) of R are surjective.Using R[M;σ],we provide various classes of both bleached and non-bleached local rings.One of the main problems concerning strongly clean rings is to characterize the rings R for which the matrix ring M_(n)(R)is strongly clean.We investigate the strong cleanness of the full matrix rings over the skew monoid ring R[M;σ].展开更多
Let R be an associative ring with identity and Z^*(K)be its set of non-zero zero-divisors.The undirected zero-divisor graph of R、denoted byΓ(R),is the graph whose vert ices are the non-zero zero-divisors of R、and w...Let R be an associative ring with identity and Z^*(K)be its set of non-zero zero-divisors.The undirected zero-divisor graph of R、denoted byΓ(R),is the graph whose vert ices are the non-zero zero-divisors of R、and where two distinct verticesγand s are adjacent if and only ifγs=0 or sγ=0.The dist ance bet ween vertices a and b is the length of the shortest path connecting them,and the diameter of the graph,diam(Γ(R)),is the superimum of these distances.In this paper,first we prove some results aboutΓ(R)of a semi-commutative ring R.Then,for a reversible ring R and a unique product monoid M、we prove 0≦diam(Γ(R))<diam(Γ(R[M]))≦3.We describe all the possibilities for the pair diam(Γ(R))and diam(Γ(R[M])),strictly in terms of the properties of a ring R,where K is a reversible ring and M is a unique product monoid.Moreover,an example showing the necessity of our assumptions is provided.展开更多
For a monoid M and an endomorphism α of a ring R, we introduce the notion of strongly M-α-reflexive rings and study its properties. For an u.p.-monoid M and a right Ore ring R with its classical right quotient ring ...For a monoid M and an endomorphism α of a ring R, we introduce the notion of strongly M-α-reflexive rings and study its properties. For an u.p.-monoid M and a right Ore ring R with its classical right quotient ring Q, we prove that R is strongly M-α-reflexive if and only if Q is strongly M-α-reflexive, where R is α-rigid, α is an epimorphism of R. The relationship between some special subrings of upper triangular matrix rings and strongly M-α-reflexive rings is also investigated. Several known results similar to strongly M-α-reversible rings are obtained.展开更多
We study the reversible properties of monoid crossed products. The new class of strongly CM-reversible rings is introduced and characterized. This class of rings is a generalization of those of strongly reversible rin...We study the reversible properties of monoid crossed products. The new class of strongly CM-reversible rings is introduced and characterized. This class of rings is a generalization of those of strongly reversible rings, skew strongly reversible rings and strongly M-reversible rings. Some well-known results on this subject are generalized and extended.展开更多
The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the co...The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the correction of errors of n=p−12 length cyclic codes(C)over quaternion integers of Quaternion Mannheim(QM)weight one up to two coordinates have considered.In continuation,the case of cyclic codes of lengths n=p−12 and 2n−1=p−2 has studied to improve the error correction efficiency.In this study,we present the decoding of cyclic codes of length n=ϕ(p)=p−1 and length 2n−1=2ϕ(p)−1=2p−3(where p is prime integer andϕis Euler phi function)over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors.Furthermore,the error correction capability and code rate tradeoff of these codes are also discussed.Thus,an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability.展开更多
Let M be a monoid. A ring R is called M-π-Armendariz if whenever a = a1g1＋ a292 ＋ …＋angn, β= b1h1 ＋ b2h2 ＋ …＋ bmhm ∈ R[M] satisfy αβ ∈ nil（R[M]）, then aibj ∈ nil（R） for all i, j. A ring R is called ...Let M be a monoid. A ring R is called M-π-Armendariz if whenever a = a1g1＋ a292 ＋ …＋angn, β= b1h1 ＋ b2h2 ＋ …＋ bmhm ∈ R[M] satisfy αβ ∈ nil（R[M]）, then aibj ∈ nil（R） for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-Tr-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil（R[M]） = nil（R）[M]. Moreover, we study the relationship between the weak zip-property （resp., weak APP-property, nilpotent p.p.-property, weak associated prime property） of a ring R and that of the monoid ring RIM] in case R is M-π-Armendariz.展开更多
文摘Let R be a ring with an endomorphismσ,F∪{0}the free monoid generated by U={u1,…,ut}with 0 added,and M a factor of F obtained by setting certain monomials in F to 0 such that M^(n)=0 for some n.Then we can form the non-semiprime skew monoid ring R[M;σ].A local ring R is called bleached if for any j∈J(R)and any u∈U(R),the abelian group endomorphisms l_(u)−r_(j) and l_(j)−r_(u) of R are surjective.Using R[M;σ],we provide various classes of both bleached and non-bleached local rings.One of the main problems concerning strongly clean rings is to characterize the rings R for which the matrix ring M_(n)(R)is strongly clean.We investigate the strong cleanness of the full matrix rings over the skew monoid ring R[M;σ].
文摘Let R be an associative ring with identity and Z^*(K)be its set of non-zero zero-divisors.The undirected zero-divisor graph of R、denoted byΓ(R),is the graph whose vert ices are the non-zero zero-divisors of R、and where two distinct verticesγand s are adjacent if and only ifγs=0 or sγ=0.The dist ance bet ween vertices a and b is the length of the shortest path connecting them,and the diameter of the graph,diam(Γ(R)),is the superimum of these distances.In this paper,first we prove some results aboutΓ(R)of a semi-commutative ring R.Then,for a reversible ring R and a unique product monoid M、we prove 0≦diam(Γ(R))<diam(Γ(R[M]))≦3.We describe all the possibilities for the pair diam(Γ(R))and diam(Γ(R[M])),strictly in terms of the properties of a ring R,where K is a reversible ring and M is a unique product monoid.Moreover,an example showing the necessity of our assumptions is provided.
基金partially supported by the Provincial Natural Science Foundation of Anhui Province of China（KJ2017A040）
文摘For a monoid M and an endomorphism α of a ring R, we introduce the notion of strongly M-α-reflexive rings and study its properties. For an u.p.-monoid M and a right Ore ring R with its classical right quotient ring Q, we prove that R is strongly M-α-reflexive if and only if Q is strongly M-α-reflexive, where R is α-rigid, α is an epimorphism of R. The relationship between some special subrings of upper triangular matrix rings and strongly M-α-reflexive rings is also investigated. Several known results similar to strongly M-α-reversible rings are obtained.
基金The NSF(11601005) of Chinathe Jiangsu Planned Projects(1601151C) for Postdoctoral Research Funds+1 种基金the Provincial NSF(KJ2017A040) of Anhui Provincethe Graduate Students Innovation Projects(2016141) of Anhui University of Technology
文摘We study the reversible properties of monoid crossed products. The new class of strongly CM-reversible rings is introduced and characterized. This class of rings is a generalization of those of strongly reversible rings, skew strongly reversible rings and strongly M-reversible rings. Some well-known results on this subject are generalized and extended.
基金The authors extend their gratitude to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number R.G.P.1/85/42.
文摘The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the correction of errors of n=p−12 length cyclic codes(C)over quaternion integers of Quaternion Mannheim(QM)weight one up to two coordinates have considered.In continuation,the case of cyclic codes of lengths n=p−12 and 2n−1=p−2 has studied to improve the error correction efficiency.In this study,we present the decoding of cyclic codes of length n=ϕ(p)=p−1 and length 2n−1=2ϕ(p)−1=2p−3(where p is prime integer andϕis Euler phi function)over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors.Furthermore,the error correction capability and code rate tradeoff of these codes are also discussed.Thus,an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability.
基金This work was supported by the National Natural Science Foundation of China （Grant No. 11071097） and the Natural Science Foundation of Jiangsu Province （BK20141476）.
文摘Let M be a monoid. A ring R is called M-π-Armendariz if whenever a = a1g1＋ a292 ＋ …＋angn, β= b1h1 ＋ b2h2 ＋ …＋ bmhm ∈ R[M] satisfy αβ ∈ nil（R[M]）, then aibj ∈ nil（R） for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-Tr-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil（R[M]） = nil（R）[M]. Moreover, we study the relationship between the weak zip-property （resp., weak APP-property, nilpotent p.p.-property, weak associated prime property） of a ring R and that of the monoid ring RIM] in case R is M-π-Armendariz.
