\((\delta,2)\)-primary ideals of a commutative ring. (English) Zbl 1513.13013

Czech. Math. J. 70, No. 4, 1079-1090 (2020); erratum ibid. 70, No. 4, 1219 (2020).
Summary: Let \(R\) be a commutative ring with nonzero identity, let \(\mathcal{I(R)}\) be the set of all ideals of \(R\) and \(\delta\colon\mathcal{I(R)}\rightarrow\mathcal{I(R)}\) an expansion of ideals of \(R\) defined by \(I\mapsto\delta(I)\). We introduce the concept of \((\delta,2)\)-primary ideals in commutative rings. A proper ideal \(I\) of \(R\) is called a \((\delta,2)\)-primary ideal if whenever \(a,b\in R\) and \(ab\in I\), then \(a^{2}\in I\) or \(b^{2}\in\delta(I)\). Our purpose is to extend the concept of \(2\)-ideals to \((\delta,2)\)-primary ideals of commutative rings. Then we investigate the basic properties of \((\delta,2)\)-primary ideals and also discuss the relations among \((\delta,2)\)-primary, \(\delta\)-primary and \(2\)-prime ideals.


13A15 Ideals and multiplicative ideal theory in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
05A15 Exact enumeration problems, generating functions
13G05 Integral domains
Full Text: DOI


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