One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summa...One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.展开更多
The two-dimensional(2D)C3 N has emerged as a material with promising applications in high performance device owing to its intrinsic bandgap and tunable electronic properties.Although there are several reports about th...The two-dimensional(2D)C3 N has emerged as a material with promising applications in high performance device owing to its intrinsic bandgap and tunable electronic properties.Although there are several reports about the bandgap tuning of C3 N via stacking or forming nanoribbon,bandgap modulation of bilayer C3 N nanoribbons(C3NNRS)with various edge structures is still far from well understood.Here,based on extensive first-principles calculations,we demonstrated the effective bandgap engineering of C3 N by cutting it into hydrogen passivated C3 NNRS and stacking them into bilayer heterostructures.It was found that armchair(AC)C3 NNRS with three types of edge structures are all semiconductors,while only zigzag(ZZ)C3NNRS with edges composed of both C and N atoms(ZZCN/CN)are semiconductors.The bandgaps of all semiconducting C3 NNRS are larger than that of C3 N nanosheet.More interestingly,AC-C3 NNRS with CN/CN edges(AC-CN/CN)possess direct bandgap while ZZ-CN/CN have indirect bandgap.Compared with the monolayer C3 NNR,the bandgaps of bilayer C3NNRS can be greatly modulated via different stacking orders and edge structures,varying from 0.43 eV for ZZ-CN/CN with AB’-stacking to 0.04 eV for AC-CN/CN with AA-stacking.Particularly,transition from direct to indirect bandgap was observed in the bilayer AC-CN/CN heterostructure with AA^stacking,and the indirect-to-direct transition was found in the bilayer ZZ-CN/CN with ABstacking.This work provides insights into the effective bandgap engineering of C3 N and offers a new opportunity for its applications in nano-electronics and optoelectronic devices.展开更多
文摘One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.
基金This work was supported by the National Natural Science Foundation of China(Grant No.21673075).
文摘The two-dimensional(2D)C3 N has emerged as a material with promising applications in high performance device owing to its intrinsic bandgap and tunable electronic properties.Although there are several reports about the bandgap tuning of C3 N via stacking or forming nanoribbon,bandgap modulation of bilayer C3 N nanoribbons(C3NNRS)with various edge structures is still far from well understood.Here,based on extensive first-principles calculations,we demonstrated the effective bandgap engineering of C3 N by cutting it into hydrogen passivated C3 NNRS and stacking them into bilayer heterostructures.It was found that armchair(AC)C3 NNRS with three types of edge structures are all semiconductors,while only zigzag(ZZ)C3NNRS with edges composed of both C and N atoms(ZZCN/CN)are semiconductors.The bandgaps of all semiconducting C3 NNRS are larger than that of C3 N nanosheet.More interestingly,AC-C3 NNRS with CN/CN edges(AC-CN/CN)possess direct bandgap while ZZ-CN/CN have indirect bandgap.Compared with the monolayer C3 NNR,the bandgaps of bilayer C3NNRS can be greatly modulated via different stacking orders and edge structures,varying from 0.43 eV for ZZ-CN/CN with AB’-stacking to 0.04 eV for AC-CN/CN with AA-stacking.Particularly,transition from direct to indirect bandgap was observed in the bilayer AC-CN/CN heterostructure with AA^stacking,and the indirect-to-direct transition was found in the bilayer ZZ-CN/CN with ABstacking.This work provides insights into the effective bandgap engineering of C3 N and offers a new opportunity for its applications in nano-electronics and optoelectronic devices.
