The aim of this paper is to study the Hopfian property in the context of chain and duo rings. For such rings, we characterize Hopfian free modules and show that a direct sum of cyclic R-modules is Hopfian if and only if the sum is finite. This allows us to show that finitely generated modules over a local right duo ring, which has the FGC-property, are Hopfian and cancel in direct sums. Moreover, being finitely, hopficity, and the cancellation property are equivalent for modules over Artinian rings.

中文翻译：

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Hopficity 和双环

本文的目的是在链和双环的背景下研究霍普夫性质。对于这样的环，我们刻画了 Hopfian 自由模，并表明循环R模的直接和是 Hopfian 当且仅当和是有限的。这使我们能够证明在具有FGC 性质的局部右双环上的有限生成模块是 Hopfian 并在直接和中取消。此外，有限性、跳跃性和取消属性对于阿蒂尼环上的模块是等效的。