## An amalgamated duplication of a ring along an ideal: the basic properties.(English)Zbl 1126.13002

Let $$R$$ be a (commutative) ring and let $$E$$ be an $$R$$-submodule of the total ring of fractions of $$R$$ that is closed under multiplication. The authors define the ring $$R\ltimes E$$ (the amalgamated duplication of the ring $$R$$ along the $$R$$-module $$E$$) as a direct sum of $$R$$-modules $$R\oplus E$$ with a naturally defined multiplication: $$(r,e)(s,f)=(rs,rf+se+ef)$$ for $$r,s\in R$$ and $$e,f\in E$$. (cf. Nagata’s idealization.)
The authors present the ring $$R\ltimes E$$ as a pullback and they use this presentation to relate properties of $$R\ltimes E$$ to properties of $$R$$ and of $$R+E$$: for example, $$R\ltimes E$$ is noetherian iff both $$R$$ and $$R+E$$ are noetherian. The authors study especially the case when $$E=I$$ is an ideal of $$R$$. They prove that $$\dim R\ltimes I=\dim R$$, that $$R\ltimes I$$ is noetherian iff $$R$$ is noetherian, etc. They also describe the prime spectrum of $$R\ltimes I$$.

### MSC:

 13A15 Ideals and multiplicative ideal theory in commutative rings 13B99 Commutative ring extensions and related topics 14A05 Relevant commutative algebra
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### References:

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