Closure formula for ideals in intermediate rings

[EN] In this paper, we prove that the closure formula for ideals in C(X) under m topology holds in intermediate ring also, i.e. for any ideal I in an intermediate ring with m topology, its closure is the intersection of all the maximal ideals containing I.Kharbhih, JPJ.; Dutta, S. (2020). Closure formula for ideals in intermediate rings. Applied General Topology. 21(2):195-200. https://doi.org/10.4995/agt.2020.11903OJS195200212S. K. Acharyya and B. Bose, A correspondence between ideals and z-filters for certain rings of continuous functions - some remarks, Topology and its Applications 160, no. 13 (2013), 1603-1605. https://doi.org/10.1016/j.topol.2013.06.011S. K. Acharyya, K. C. Chattopadhyay and D. P. Ghosh, A class of subalgebras of C(X) and the associated compactness, Kyungpook Math. J. 41, no. 2 (2001), 323-324.S. K. Acharyya and D. De, An interesting class of ideals in subalgebras of C(X) containing C*(X), Comment. Math. Univ. Carolin. 48, no. 2 (2007), 273-280.S. K. Acharyya and D. De, Characterization of function rings between C*(X) and C(X), Kyungpook Math. J. 46 (2006), 503-507.H. L. Byun and S. Watson, Prime and maximal ideals in subrings of C(X), Topology and its Applications 40 (1991), 45-62. https://doi.org/10.1016/0166-8641(91)90057-SJ. M. Domínguez and J.-Gómez Pérez, Intersections of maximal ideals in algebras between C*(X) and C(X), Topology and its Applications 98 (1999), 149-165. https://doi.org/10.1016/S0166-8641(99)00043-7L. Gillman, M. Henriksen and M. Jerison, On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. 5 (1954), 447-455. https://doi.org/10.1090/S0002-9939-1954-0066627-6L. Gillman and M. Jerison, Rings of continuous functions, Univ. Ser. Higher Math, D. Van Nostrand Company, Inc., Princeton, N. J., 1960. https://doi.org/10.1007/978-1-4615-7819-2E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64, no. 1 (1948), 45-99. https://doi.org/10.1090/S0002-9947-1948-0026239-9D. Plank, On a class of subalgebras of C(X) with applications to $beta X setminus X$, Fund. Math. 64 (1969), 41-54. https://doi.org/10.4064/fm-64-1-41-54L. Redlin and S. Watson, Maximal ideals in subalgebras of C(X), Proc. Amer. Math. Soc. 100, no. 4 (1987), 763-766. https://doi.org/10.2307/2046719T. Shirota, On ideals in rings of continuous functions, Proc. Japan Acad. 30, no. 2 (1954), 85-89. https://doi.org/10.3792/pja/119552617

On annihilator graph of a finite commutative ring

‎The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $neq$ $ann(xy)$‎. ‎In this paper we give the sufficient condition for a graph $AG(R)$ to be complete‎. ‎We characterize rings for which $AG(R)$ is a regular graph‎, ‎we show that $gamma (AG(R))in {1,2}$ and we also characterize the rings for which $AG(R)$ has a cut vertex‎. ‎Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph‎

Topological indices of total graph of the ring Zո × Zm

In this paper, we compute some distance-based topological indices namely Wiener index, hyper Wiener index and reverse Wiener index of the total graph of the ring Zn × Zm. We also compute some eccentricity-based topological indices namely first Zagreb eccentricity index, second Zagreb eccentricity index, eccentric connectivity index, connective eccentricity index and eccentric distance sum index of the total graph of the ring Zn × Zm. Finally, we compute some degree-based topological indices namely second Zagreb index, product-connectivity index, sum connectivity index, atom-bond connectivity index and geometric arithmetic index of the total graph of the ring Zn ×Zm when both n and m are even, when n or m is even, and the case when n and m are both prime numbers (not neccesarily distinct).Publisher's Versio