Morita系统环上的一致模和Hollow模
|
摘要
利用Morita系统环上(右)模的分解,讨论其上模的本质子模和多余子模的结构.对于Morita系统环■,每个右T-模都可以分解为一个四元对(P,Q)_(f,g),给出其上的一致模和hollow模的结构刻画,并给出(P,Q)_(f,g)是一致(hollow)模的必要条件.记L={p∈P g(p■m)=0,■m∈M},K={q∈Q f(q■n)=0,■n∈N},证明:1)若P=0,且K=Q是一致模(或Q=0,且P=L是一致模),则(P,Q)_(f,g)是一致模;2)若P和Q是hollow模,且f(Q■N)=P,g(P■M)≠Q(或f(Q■N)≠P,g(P■M)=Q),则(P,Q)(f,g)是hollow模.
By using the decomposition of the(right)modules over rings of Morita conexts,we discussed the structures of essential submodules and small submodules of modules over these rings.For the Morita conexts ■,every right T-module could be decomposed into a quadriad(P,Q)(f,g),we gave characterizations of the structure of uniform modules and hollow modules over these rings and the necessary conditions for(P,Q)(f,g)to be uniform(hollow). Let L={p∈P g(p■m)=0,■m∈M},K={q∈Q f(q■n)=0,■n∈N},we prove that:1)If P=0,and K=Qis uniform module(or Q=0,and P=Lis uniform module),then(P,Q)(f,g)is uniform module;2)If P and Q are hollow modules,f(Q■N)=P,g(P■M)≠Q(or f(Q■N)≠P,g(P■M)=Q),then(P,Q)(f,g)is hollow module.
By using the decomposition of the(right)modules over rings of Morita conexts,we discussed the structures of essential submodules and small submodules of modules over these rings.For the Morita conexts ■,every right T-module could be decomposed into a quadriad(P,Q)(f,g),we gave characterizations of the structure of uniform modules and hollow modules over these rings and the necessary conditions for(P,Q)(f,g)to be uniform(hollow). Let L={p∈P g(p■m)=0,■m∈M},K={q∈Q f(q■n)=0,■n∈N},we prove that:1)If P=0,and K=Qis uniform module(or Q=0,and P=Lis uniform module),then(P,Q)(f,g)is uniform module;2)If P and Q are hollow modules,f(Q■N)=P,g(P■M)≠Q(or f(Q■N)≠P,g(P■M)=Q),then(P,Q)(f,g)is hollow module.
引文
[1]HAGHANY A,VARADARAJAN K.Study of Modules over Formal Triangular Matrix Rings[J].J Pure Appl Algebra,2000,147(1):41-58.
[2]HAGHANY A,VARADARAJAN K.Study of Formal Triangular Matrix Rings[J].Comm Algebra,1999,27(11):5507-5525.
[3]ENOCHS E E,TORRECILLAS B.Flat Covers over Formal Triangular Matrix Rings and Minimal Quillen Factorizations[J].Forum Math,2011,23(3):611-624.
[4]ENOCHS E E,CORTS-IZURDIAGA M,TORRECILLAS B.Gorenstein Conditions over Triangular Matrix Rings[J].J Pure Appl Algebra,2014,218(8):1544-1554.
[5]LU Ming.Gorenstein Defect Categories of Triangular Matrix Algebras[J].J Algebra,2017,480:346-367.
[6]ENOCHS E E.Injective and Flat Covers,Envelopes and Resolvents[J].Israel J Math,1981,39(3):189-209.
[7]McCONNELL J C,ROBSON J C.Noncommutative Noetherian Rings[M].New York:John Wiley and Sons,1987:12-13.
[8]GREEN E L.On the Representation Theory of Rings in Matrix Form[J].Pacific J Math,1982,100(1):123-138.
[9]ANDERSON F W,FULLER K R.Rings and Categories of Modules[M].2nd ed.New York:Springer-Verlag,1992.
[10]KESKIN D.On Lifting Modules[J].Comm Algebra,2000,28(7):3427-3440.
[2]HAGHANY A,VARADARAJAN K.Study of Formal Triangular Matrix Rings[J].Comm Algebra,1999,27(11):5507-5525.
[3]ENOCHS E E,TORRECILLAS B.Flat Covers over Formal Triangular Matrix Rings and Minimal Quillen Factorizations[J].Forum Math,2011,23(3):611-624.
[4]ENOCHS E E,CORTS-IZURDIAGA M,TORRECILLAS B.Gorenstein Conditions over Triangular Matrix Rings[J].J Pure Appl Algebra,2014,218(8):1544-1554.
[5]LU Ming.Gorenstein Defect Categories of Triangular Matrix Algebras[J].J Algebra,2017,480:346-367.
[6]ENOCHS E E.Injective and Flat Covers,Envelopes and Resolvents[J].Israel J Math,1981,39(3):189-209.
[7]McCONNELL J C,ROBSON J C.Noncommutative Noetherian Rings[M].New York:John Wiley and Sons,1987:12-13.
[8]GREEN E L.On the Representation Theory of Rings in Matrix Form[J].Pacific J Math,1982,100(1):123-138.
[9]ANDERSON F W,FULLER K R.Rings and Categories of Modules[M].2nd ed.New York:Springer-Verlag,1992.
[10]KESKIN D.On Lifting Modules[J].Comm Algebra,2000,28(7):3427-3440.