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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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