摘要
利用Morita系统环上的 (右 )模的分解 ,研究其上的自由模 ,并利用所得的结果刻画形式三角矩阵环上的自由模与投射模 .对于Morita系统环T=RMNS(θ,ψ),每个T模可以分解为一个四元素对 (P ,Q) (f,g) .记 PR =P/Imf, QS =Q/Img , R =R/Imθ, S =S/Imψ ,且设Λ为任意非空集合 ,主要结果有 :1 )若 (P ,Q) (f,g) T(Λ) ,则 P R R(Λ) , Q S S(Λ) .2 )若 1 P Rθ=0且 1 Q Sψ=0 ,则 {(pλ,qλ)λ∈Λ}是 (P ,Q) (f,g) 的一组自由基当且仅当下列条件①和②成立 :① { pλ λ∈Λ}和 { qλ λ∈Λ}分别为 P R 和 Q S 的自由基 ,且 {pλ λ∈Λ}是R线性无关的 ,qλ λ∈Λ是S线性无关的 ;②f ∑λqλ nλ =0蕴涵nλ =0 ,且g ∑λpλ mλ =0蕴涵mλ =0 (对于任意的nλ ∈N ,mλ ∈M ,λ∈Λ) .3)当M =0时 ,(P ,Q) (f,g) T(Λ) 当且仅当 P R R(Λ) , Q S S(Λ)
By the decomposition of the (right) modules over rings of Morita contexts, the authors studied free modules over these rings, characterized free modules and projective modules over formal triangular matrix rings with the results gotten already. For the ring of Morita contexts T=RM NS (θ,ψ) , every right T-module can be decomposed into a quadriad (P,Q) (f,g) . Write R=P/ Im f, S=Q/ Im g, =R/ Im θ, =S/ Im ψ , and let Λ be an arbitrary nonempty set, our main results read as follows: 1) If (P,Q) (f, g) T (Λ) , then (Λ) , (Λ) . 2) If 1 P? 璕θ=0 and 1 Q? 璖ψ=0 , then {(p λ,q λ)λ∈Λ} is a free basis of (P, Q) (f, g) if and only if the following conditions ① and ② hold: ① { λλ∈Λ} and { λλ∈Λ} are free basises of and respectively, {p λλ∈Λ} is R -linear independent and {q λλ∈Λ} is S -linear independent; ② f∑λq λn λ=0 implies n λ=0, and g∑λp λm λ=0 implies m λ=0 (for all n λ∈N, m λ∈M,λ∈Λ) . 3) When M=0, (P, Q) (f, g) T (Λ) if and only if (Λ) , (Λ) and f is a monomorphism.
出处
《东南大学学报（自然科学版）》
EI
CAS
CSCD
北大核心
2001年第5期140-145,共6页
Journal of Southeast University：Natural Science Edition
基金
国家自然科学基金资助项目 ( 1970 10 0 8)
