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Idealization of a module. (English) Zbl 1194.13002

Summary: Authors’ abstract: Let \(R\) be a commutative ring and \(M\) an \(R\)-module. Nagata introduced the idealization \(R (+) M\) of \(M\). Here \(R (+) M = R \oplus M\) (direct sum) is a commutative ring with product \((r_1 , m_1 )(r_2 , m_2 ) = (r_1 r_2 , r_1 m_2 + r_2 m_1 )\). The name comes from the fact that if \(N\) is a submodule of \(M\), then \(0 \oplus N\) is an ideal of \(R (+) M\). The idealization can be used to extend results about ideals to modules and to provide interesting examples of commutative rings with zero divisors. We survey known results concerning \(R (+) M\) and give some new ones too. The theme throughout is how properties of \(R (+) M\) are related to those of \(R\) and \(M\).

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13B99 Commutative ring extensions and related topics
13C99 Theory of modules and ideals in commutative rings
13A02 Graded rings
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