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**Relative \(z\)-ideals in commutative rings.**
*(English)*
Zbl 1264.13003

This paper deals with the fundamental properties of relative \(z\)-ideal (briefly re\(z\)-ideal) which has been introduced by Aliabad and Azarpanah and Taherifar. This class of ideals is a special kind of generalized form of the class of \(z\)-ideals. After giving a clear and informative introduction, the authors not only characterize re\(z\)-ideals but also offer some examples of re\(z\)-ideal in Section 2. Among others, they show that whenever an ideal \(I\) is a re\(z\)-ideal, then it is also a \(z_K\)-ideal for some ideal \(K\) containing \(I\) properly. In Proposition 2.8, they show that if \(I\) is an ideal in a semisimple ring \(R\) with Ann(\(I)\neq (0)\), then \(I\) is a re\(z\)-ideal. Moreover, they show that if \(P\) and \(Q\) are prime ideals in a ring \(R\) and if \(P \cap Q\) is re\(z\)-ideal, then either \(P\) or \(Q\) is a \(z\)-ideal (see Proposition 2.10). In Section 3, the authors consider large and small re\(z\)-ideals. More specifically, maximal and minimal \(z\)-factors of \(I\) are investigated for an ideal \(J\) in a ring \(R\), maximal \(z_J\)-ideals in \(R\) and for a given ideal \(I\) in \(R\). Indeed, nontrivial maximal \(z_J\)-ideals are characterized as the maximal elements of the set of prime \(z\)-ideals that do not contain \(J\) and nontrivial maximal \(z_J\)-ideals contained in \(J\) are precisely the intersection of such prime \(z\)-ideals with \(J\). The authors characterize the largest \(z\)-factor of ideals if it exists. Furthermore, they show that the largest \(z\)-factor of a semiprime re\(z\)-ideal \(I\) is the intersection of those minimal prime ideals over \(I\) which are not \(z\)-ideals. Section 4 is dedicated to the investigation of the ring \(C(X)\) via re\(z\)-ideals by which they obtain some nice results concerning the algebraic properties of \(C(X)\) and topological properties of \(X\). Among others, they show that the sum of a re\(z\)-ideal and a convex (absolutely convex) ideal is convex (absolutely convex) iff \(X\) is an \(F\)-space.

Reviewer: Saeid Jafari (Copenhagen)

### MSC:

13A15 | Ideals and multiplicative ideal theory in commutative rings |

54C40 | Algebraic properties of function spaces in general topology |

### Keywords:

convex and absolutely convex ideal; \(z\)-ideal; \(F\)-space; re\(z\)-ideal; \(z\)-factor; \(z_J\)-ideal
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\textit{A. R. Aliabad} et al., Commun. Algebra 41, No. 1, 325--341 (2013; Zbl 1264.13003)

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