On lattices of $z$-ideals of function rings

Year, pages: 2018, 271 - 284

An ideal $I$ of a ring $A$ is a $z$-ideal if whenever $a,b\in A$ belong to the same maximal ideals of $A$ and $a\in I$, then $b\in I$ as well. On the other hand, an ideal $J$ of $A$ is a $d$-ideal if $\Ann²(a)\subseteq J$ for every $a\in J$. It is known that the lattices $\mathsf{Z}(L)$ and $\mathsf{D}(L)$ of the ring $\mathcal{R}L$ of continuous real-valued functions on a frame $L$, consisting of $z$-ideals and $d$-ideals of $\mathcal{R}L$, respectively, are coherent frames. In this paper we characterize, in terms of the frame-theoretic properties of $L$ (and, in some cases, the algebraic properties of the ring $\mathcal{R}L$), those $L$ for which $\mathsf{Z}(L)$ and $\mathsf{D}(L)$ satisfy the various regularity conditions on algebraic frames introduced by Mart\'inez and Zenk [MarZen]. Every frame homomorphism $h\colon L\to M$ induces a coherent map $\mathsf{Z}(h)\colon\mathsf{Z}(L)\to\mathsf{Z}(M)$. Conditions are given of when this map is closed, or weakly closed in the sense Mart\'inez [MarIV]. The case of openness of this map was discussed in [DubIgh3]. We also prove that, as in the case of the ring $C(X)$, the sum of two $z$-ideals of $\mathcal{R}L$ is a $z$-ideal.

Dube, T., Ighedo, O. 2018. On lattices of $z$-ideals of function rings. In Mathematica Slovaca, vol. 68, no.2, pp. 271-284. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0099

Dube, T., Ighedo, O. (2018). On lattices of $z$-ideals of function rings. Mathematica Slovaca, 68(2), 271-284. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0099