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

Ulucak, Gülşen; Çelikel, Ece Yetkin

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.

MSC:

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

Keywords:

\((\delta,2)\)-primary ideal; \(2\)-prime ideal; \(\delta\)-primary ideal

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