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 当且仅当 The question that how to determine a module M over a commutative ring R with invariant factors was discussed,and the result indicated that M was such an R-module if and only if the fractional module S-1M over the fractional ring S-1R had invariant factors whenever the multiplicative subset S of R was invertible in R/AnnR(M) being the annihilator of M in R. 探讨了交换环R上具有不变因子的模M之判别问题,证明了只要R的乘法子集S在R/AnnR(M)中可逆,则M为具有不变因子的R-模当且仅当分式模S-1M为分式环S-1R上的具有不变因子的模. 2. N is △( M )-injective if and only if N is △( Mn)-injective. ②N是△(M)-内射模当且仅当N是△(Mn)-内射模. In this paper we prove that the equation xn+yn=zφ(n) has positive integer solutions(x,y,z) with gcd(x,y)=1 if and only if it is n≤3. 证明了方程xn+yn=zφ(n)当且仅当n≤3时有正整数解(x,y,z)适合gcd(x,y)=1. It is proved that the ring [[Rs,≤,λ]] is a reduced left PP-ring if and only if R is a reduced left PP-ring and every S-indexed subset C of B(R) has a least upper bound in B(R); 证明了[[Rs,≤,λ]]是reduced左PP-环当且仅当R是reduced左PP-环,且B(R)的每个S可标子集C在B(R)中有最小上界; R is a ring without nonzero zero-divisors,then the ring [[Rs,≤,λ]] is Dedeking finite if and only if R is Dedeking finite. 若环R无非零零因子,则[[Rs,≤,λ]]是Dedekind有限环当且仅当R是Dedeking有限环. G(+-+) and G~(-+-) are regular if and only if G≌C_5 or K_7 or K_2 or K_(3,3) or G_0; G~(+-+)和G~(-+-)是正则图当且仅当G为C_5、K_7、K_2、K_(3,3)或G_0; G~(-++) and G~(+--) are regular if and only if G is (n-1)/2-regular. We also give some upper bounds on the spectral radius of transformation graphs. Then we estimate these upper bounds. G~(-++)和G~(+--)是正则的当且仅当G是(n-1)/2-正则图．同时还讨论了变换图的谱半径上界,并对这些上界进行了估计． Let S be an antinegative commutative semiring and Mn(S) be the semiring of all n×n matrices over S.For a linear operator L on Mn(S),L strongly preserves invertible matrices in Mn(S) if for any A∈ Mn(S) A is invertible if and only if L(A) is invertible. 设S是一非负交换半环,Mn(S)是S上所有矩阵构成的半环. 对Mn(S)上一线性算子L,如果对任何A∈Mn(S),A可逆当且仅当L(A)可逆,则称L强保持Mn(S)中的可逆矩阵. :Assume f∈BMO1(B),then Tf is bounded on L2a(B) if and only if f~ is bounded; 则Tf在L2a(B)上有界当且仅当f~有界; Tf is compact on L2a(B) if and only if f~(z)→0(z→B). Tf在La2(B)上是紧的当且仅当~f(z)→0(z→B)。 It is proved that a wrpp semigroup satisfies a permutation identity if and only if it satisfies the identity xyzw=xzyw. 证明了wrpp半群满足置换恒等式当且仅当它满足恒等式xyzw=xzyw. Consequently,it is shown that R is S×T-weak Armendariz if and only if R is both S-weak Armendariz and T-weak Armendariz. 进而,得到了R是S×T-弱Armendariz环当且仅当R既是S-弱Armendari环又是T-弱Armendariz环. Finally,it is shown that R1×R2 is S-weak Armendariz if and only if both R1 and R2 are S-weak Armendariz. 最后,得到R1×R2是S-弱Armendariz环当且仅当R1和R2均为S-弱Armendariz环. In this paper,the sharp upper bound on the spectral radius of Nordhaus-Gaddum type at trees is given. The paper shows thatρ(T)+ρ(T~c)≤■+n-2,the equality holds if and only if T≌K_(1,n-1). 给出了n阶树的Nordhaus-Gaddum类型谱半径即图及其补图的谱半径之和的可达上界:ρ(T)+ρ(Tc)≤■+n-2,等号成立当且仅当T K1,n-1,其中Tc为T的补图,K1,n-1为n阶星图. The results show that Ti(i≤3) separation properties are good R(L)-extension,i. e. (LX,δ) is Ti spaces if and only if induced R(L)-topological space(R(L)X,ω(δ)) is Ti spaces correspondingly. 研究表明:Ti(i≤3)分离性是R(L)好的推广,即(LX,δ)是Ti空间当且仅当其R(L)型诱导空间(R(L)X,ω(δ))也是Ti空间. Let V∈｛Sn(F),Mn(F)｝,a map Φ:V→V is said to preserve idempotence if A-λB is idempotent if and only if Φ(A)-λΦ(B) is idempotent for any A,B∈V and λ∈F. 设V∈{Sn(F),Mn(F)},对任意的A,B∈V和λ∈F,如果A-λB幂等当且仅当Φ(A)-λΦ(B)幂等,则称映射Φ:V→V是保幂等性的. When the characteristic of F is 0,it is shown that Φ:Sn(F)→Sn(F) is a map preserving idempotence if and only if there exists an invertible matrix P∈Mn(F) with PtP=aIn for some nonzero scalar a in F such that Φ(A)=PAP-1 for every A∈Sn(F). 证明了:如果F的特征为0,Φ:Sn(F)→Sn(F),则Φ是一个保幂性映射当且仅当存在Mn(F)中的一个可逆阵P使得对Sn(F)中的每一个A都有Φ(A)=PAP-1,其中P满足PtP=aIn,a为F中的一个非零元. We also proved S∈ +S,S is multiplicative band semirings if and only if S is distributive lattice. 证明了如果半环S的加法半群是半格,则S是乘法带半环当且仅当S是分配格; (3) A space X is nearly metacompact if and only if every monotone open cover U has an open refinement that is point-finite on some dense subset of X; (3)空间X是几乎亚紧的当且仅当X的每一单调开覆盖U有一个开加细,且在X的某一稠密子集上是点有限的; Let f be a surjection and (Xf,df) be the inverse limit space generated by (X,f),and σf be the shift map on Xf. then f has PWPOTP(PWPOTP+)if and only if σf has PWPOTP(PWPOTP+); 满射f有PWPOTP(PWPOTP+)当且仅当由(X,f)生成的逆极限系统(Xf,σf)上转移自映射σf有PWPOTP(PWPOTP+);

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