Let |$K$| be a commutative Noetherian ring with identity, let |$A$| be a |$K$|-algebra and let |$B$| be a subalgebra of |$A$| such that |$A/B$| is finitely generated as a |$K$|-module. The main result of the paper is that |$A$| is finitely presented (resp. finitely generated) if and only if |$B$| is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on |$K$| and show that for finite generation it can be replaced by a weaker condition that the module |$A/B$| be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension |$1$| of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

中文翻译：

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有限CO-Rank的子环和子代数的演示

令| $ K $ | 是具有身份的可交换的Noether环，让| $ A $ | 成为| $ K $ | -代数并让| $ B $ | 是| $ A $ |的子代数 这样| $ A / B $ | 是作为| $ K $ |有限生成的 -模块。本文的主要结果是| $ A $ | 当且仅当| $ B $ |时才有限地呈现（分别有限生成）是有限表示的（分别是有限生成的）。作为推论，我们得到：有限表示环中有限索引的子环是有限表示的；场上有限表示的代数中有限维次子代数是有限表示的（已经由Voden于2009年展示）。我们还将讨论Noetherian假设对| $ K $ |的作用。并表明，对于有限生成，可以用| $ A / B $ |模块的弱条件代替它。有限地呈现。最后，我们通过表现出理想的维数| $ 1 $ |，证明了结果不容易扩展到非缔合代数。并非有限生成为李代数的秩2的自由李代数的。