Richart模

RICHART MODULES

本文引入左Richart模的概念．设M是左R模,若EndR(M)中任意元φ在M中的左零化子是M的直和项,则称M是左Richart模．左Richart模是左Richart环的推广．在文章中我们给出了左Richart环和左Richart模的等价刻画条件．探讨了Baer模和左Richart模的关系及左Richart模的性质:Baer模是左Richart模,而左Richart模不一定是Baer模;左Richart模的直和项是左Richart模,但左Richart模的直和不一定是左Richart模,我们给出了左Richart模对直和封闭的等价条件;并且证明了有限生成的Abel群是左Richart模当且仅当它是半单模或无挠模．此外,我们还探讨了左Richart模与一些重要的环、模类之间的关系,得到了左Richart模的自同态环是左Richart环,以及左Richart环的中心是VN-正则环．特别地,当模的自同态环是交换环时,模是左Richart模当且仅当它的自同态环是VN-正则环．

We introduce the notion of left Richart property in a general module theoretic setting. Suppose M is a left R _ module. If the left annihilator of any element of EndR（M） is a direct summand of M, then M is called left Richart. Left Richart modules are the extension of left Richart rings. In this paper, we give the equivalent characterizations of left Richart rings and left Richart modules, and show that a direct summand of a left Richart module inherits the property and that every finitely generated Abelian group is left Richart exactly if and only if it is either semisimple or torsion-free. Among other results, a sufficient and necessary condition for a sum of left Richart modules to be left Richart is provided. We show that the endomorphism ring of a left Richart module is a left Richart ring, and the center of a left Richart ring is reguiar. In particular, if the endomorphism ring of a module is communicative, then the module is left Richart if and only if the endomorphism ring is VN-regular ring.