关于全不变子模的两个定理的推广

On Generalizations of Two Theorms Concerning Fully Invariant Submodules

对全不变子模的两个定理:1.设M是右R-模,M=M1 M2,若N≤SMR,那么N=N1 N2,其中Ni=N∩Mi≤S(Mi)R,i=1,2;2.设M是右R-模,M=M1 M2,若F1≤S(M1)R,那么存在F2≤S(M2)R,使得F1 F2≤SMR.进行推广,则为:1'.设M是右R-模,M= i∈ΛMi,若N≤SMR,那么N= i∈ΛNi,其中Ni=N∩Mi≤S(Mi)R,i∈Λ;2'.设M是右R-模,M= i∈ΛMi,若F1≤S(M1)R,那么存在Fi≤S(Mi)R,i∈Λ-{1},使得 i∈ΛFi≤SMR.

Two theorems concerning fully invariant submodules: 1, Let M be a right R-module, and let M=M_1M_2, be a direct sum decomposition. If N≤_SM_R, then N=N_1N_2, where N_i=N∩M_i≤_S(M_i)_R, i=1, 2; and 2, Let M be a right R-module, with M=M_1M_2, and let F_1≤_S(M_1)_R. Then there exists F_2≤_S(M_2)_R, so that F_1F_2≤_SM_R are generalized as:1',Let M be a right R-module and let M=_(i∈Λ)M_i bea direct sum decomposition, for an index i∈Λ. If N≤_SM_R, then N=_(i∈Λ)N_i, where N_i=N∩M_i≤_S(M_i)_R, i∈Λ; and 2', Let M be a right R-module , with M=_(i∈Λ)M_i ,for an index set Λ, and let F_1≤_S(M_1)_R, Then there exists F_i≤_S(M_i)_R,i∈Λ-{1}, so that (i∈ΛFi≤_SM_R.