Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct...Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.展开更多
In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V ...In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumptions on A, such as commutativity of A, we prove that an A-module V has finitely many submodules if and only if V can be written as a direct sum of a cyclic A-module having only finitely many A-submodules and a finite A-module.展开更多
文摘Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.
文摘In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumptions on A, such as commutativity of A, we prove that an A-module V has finitely many submodules if and only if V can be written as a direct sum of a cyclic A-module having only finitely many A-submodules and a finite A-module.
