On \(n\)-absorbing ideals and \((m, n)\)-closed ideals in trivial ring extensions of commutative rings. (English) Zbl 1448.13006
For a commutative ring \(A\) and an \(A\)-module \(M\), the author study the behaviour of \(n\)-absorving, and strongly \(n\)-absorving ideals in the trivial extensions \(A(+)M\). In particular, for any ideal \(I\subseteq{A}\) and any submodule \(N\subseteq{M}\) such that \(IM\subseteq{N}\) they obtain that \(I\subseteq{A}\) is \(n\)-absorving iff \(I(+)N\subseteq{A(+){M}}\) is \(n\) absorving. After that, they study the case of an integral domain \(A\) with field of fractions \(K\), a \(K\)-module \(M\), and the trivial ring extension \(A(+)M\), and show that the conjecture one (resp. three) in [D. F. Anderson and A. Badawi, Commun. Algebra 39, No. 5, 1646–1672 (2011; Zbl 1232.13001)] holds in \(A(+)M\) iff it holds in \(A\) (Theorems 3.4 and 4.2 respectively). Particular cases of integral domains, as Prüfer, noetherian and \(u\)-rings are considered.
The second part of the paper deals with \((m,n)\)-ideals in trivial ring extensions.
The second part of the paper deals with \((m,n)\)-ideals in trivial ring extensions.
Reviewer: Pascual Jara (Granada)
MSC:
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
| 13G05 | Integral domains |
Keywords:
prime ideal; radical ideal; 2-absorbing ideal; \(n\)-absorbing ideal \((m, n)\)-closed ideal; trivial extension; idealization of a ringCitations:
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