Residual finiteness and the Hopf property in rings
Residual finiteness and the Hopf property in rings
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DOI：
10.1016/0021-8693(70)90087-6
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发表时间：
1970-05
影响因子：
0.9
通讯作者：
M. Orzech;L. Ribes
中科院分区：
文献类型：
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作者：
M. Orzech;L. Ribes
一个结合环(分别称为群)称为剩余有限的，如果对每个非零(分别为不恒等)元x存在一个不包含x的双边理想(分别为正规子群)，且使得剩余类环(分别为群)是有限的。剩余有限群和环似乎具有类似的性质。例如，众所周知自由群是剩余有限的([5]，第414页)，相应地，自由环d也是剩余有限的。(J·莱文告诉我们的这一结果似乎是未发表的，我们在第三节中给出了它的一个证明。)在第二节中，我们证明了如果A是合适的环(例如，Z)，则有限生成的A-代数是剩余有限的；群的相应事实是Abelian群的基本定理的结果。在第四节中，我们证明了J.Lewin定理的一个锐化版本，即有限生成的剩余有限环是Hopfian的(即，它不允许真的关于自同态)。证明了有限生成的交换A-代数关于A-代数映射是Hopfian的，并利用类似的技巧构造了Vasconelos定理的一个新的证明。在第五节中，我们指出了一些关于群代数是Hopfian或余Hopfian的判据。R3-代数
An associative ring (respectively, a group) is said to be residually finite if for each nonzero (respectively, nonidentity) element x there is a two-sided ideal (respectively, normal subgroup) not containing x and such that, the residue class ring (respectively, group) is finite. Residually finite groups and rings seem to have analogous properties. For example, it is known that free groups are residually finite ([5], p. 414) an correspondingly, that free rings d arc residually finite.(This result, communicated to us by J. Lewin, appears to be unpublished, and we have included a proof of it in Section 3.) In Section 2 we prove that if A is a suitable ring (eg, Z), then a finitely generated A-algebra is residually finite; the corresponding fact for groups is a consequence of the Fundamental Theorem for Abelian groups. In Section 4 we prove a sharpened version of J. Lewin’s theorem that a finitely generated residually finite ring is Hopfian (ie, it admits no proper onto endomorphisms). It is shown that a finitely generated commutative A-algebra is Hopfian with respect to A-algebra maps, and using a similar technique, a new proof is constructed for a theorem of Vasconcelos. In Section 5 some criteria are noted which imply that a group algebra is Hopfian or co-Hopfian.2. r3-ALGEBRAS