When a (dual-)Baer module is a direct sum of (co-)prime modules

¨¨—˚¨¯ ¸¯˚—˛˝˝¯, ¯¨¯˚¨¯ ¨˙´¯¨, N. Ghaedan
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Abstract

Since 2004, Baer modules have been considered by many authors as a generalization of the Baer rings. A module $M_R$ is called Baer if every intersection of the kernels of endomorphisms on $M_R$ is a direct summand of $M_R$. It is known that commutative Baer rings are reduced. We prove that if a Baer module M is a direct sum of prime modules, then every direct summand of M is retractable. The converse is true whenever the triangulating dimension of $M$ is finite (e.g. if the uniform dimension of M is finite). Dually, if every direct summand of a dual-Baer module M is co-retractable, then it is a direct sum of co-prime modules and the converse is true whenever the sum is finite or M is a max-module. Among other applications, we show that if R is a commutative hereditary Noetherian ring then a finitely generated R-module is Baer iff it is projective or semisimple. Also, over a ring Morita equivalent to a perfect duo ring, all dual-Baer modules are semisimple.
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当(对偶-)Baer模是(协-)素模的直和时
自2004年以来,许多作者认为贝尔模是贝尔环的推广。如果$M_R$上的自同态核的每个交集都是$M_R$M的直和数,则模$M_R$2称为Baer。已知交换Baer环是约化的。我们证明了如果Baer模M是素数模的直和,那么M的每个直和都是可伸缩的。只要$M$的三角化维数是有限的(例如,如果M的一致维数是有限),则反之亦然。对偶上,如果对偶Baer模M的每个直和都是可共伸缩的,则它是共素模的直和,并且当和是有限的或M是最大模时,反之亦然。在其他应用中,我们证明了如果R是交换的遗传诺瑟环,那么有限生成的R模是Baer,当它是射影或半单的。此外,在一个等价于完全对偶环的环Morita上,所有的对偶Baer模都是半单的。
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来源期刊
CiteScore
1.00
自引率
25.00%
发文量
15
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