P. Mangamma; R. V. G. Ravi Kumar

The Spectral Topology of Idealistic S-Algebras: Functoriality, Minimal Primes, and Irreducible Components

PDF (761KB), PP.81-100

Views: 0 Downloads: 0

Author(s)

P. Mangamma ^1,2,* R. V. G. Ravi Kumar ³

1. Department of Mathematics, JNTU Kakinada, India

2. Department of Mathematics, Government Degree College, Marripalem, ASR District, India

3. Department of Mathematics, GVP College of Engineering(A), Visakhapatnam, AP, India

* Corresponding author.

DOI: https://doi.org/10.5815/ijmsc.2026.02.06

Received: 28 Jan. 2026 / Revised: 17 Mar. 2026 / Accepted: 20 Apr. 2026 / Published: 8 Jun. 2026

Index Terms

Spectral Spaces, Hull–Kernel Topology, Prime Ideals, Sobriety, Minimal Primes, Irreducible Components, S-Algebras, Spectral Duality, Non-Hausdorff Topology

Abstract

We study the prime spectrum of idealistic S-algebras, defined via an algebraic structure with a complete lattice of ideals and a suitable notion of prime ideals. The spectrum is equipped with a natural topology and is shown to form a spectral space, possessing key properties such as compactness, separation, and sobriety. We further establish that the spectrum construction is functorial and provides a correspondence between minimal prime ideals and irreducible components under appropriate conditions. Examples are included to illustrate the role of the underlying structure, showing that the existence of minimal primes depends critically on the ideal-theoretic properties.

Cite This Paper

P. Mangamma, R. V. G. Ravi Kumar, "The Spectral Topology of Idealistic S-Algebras: Functoriality, Minimal Primes, and Irreducible Components", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.12, No.2, pp. 81-100, 2026. DOI: 10.5815/ijmsc.2026.02.06

Reference

[1]S. Abramsky and A. Jung, Domain theory, in Handbook of Logic in Computer Science, Vol. 3, Oxford Univ. Press, 1994. (ISBN: 9780198537625)
[2]R. Balbes and P. Dwinger, Distributive Lattices, Univ. of Missouri Press, 1974. (ISBN: 0826201633; Review DOI: 10.2307/2271885)
[3]R. Bandaru, R. Sirisetti, S. R. Kola, R. Noorbhasha, H. Bordbar, and A. Iampan, "Stone paradistributive latticoids," European Journal of Pure and Applied Mathematics 18 (2025), no. 4, Article 7135. DOI: 10.29020/nybg.ejpam.v18i4.7135
[4]N. A. Bazhenov, I. Sh. Kalimullin, and M. V. Schwidefsky, "An effective version of the Stone duality," (2026). arXiv: 2604.04492
[5]C. Berger, "Stone duality for spectral sheaves and the patch monad," Journal of Pure and Applied Algebra 227 (2023), 107306. DOI: 10.1016/j.jpaa.2023.107306
[6]G. Bezhanishvili, L. Carai, and P. Morandi, "Duality theory for bounded lattices: A comparative study," (2025). arXiv: 2502.21307
[7]Craig, M. Haviar, and J. S. João, "Dual spaces of lattices and semidistributive lattices," (2025). arXiv: 2509.04417
[8]Grothendieck, Éléments de géométrie algébrique: I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960). DOI: 10.1007/BF02684778
[9]M. Hochster, "Prime ideal structure in commutative rings," Transactions of the American Mathematical Society 142 (1969), 43–60. DOI: 10.1090/S0002-9947-1969-0251026-X
[10]H. Hou and A. Shen, "Stone duality of Lawson compact algebraic L-domains," (2026). arXiv: 2602.12861
[11]Z. Izhakian and L. Rowen, "Supertropical algebra," Advances in Mathematics 225 (2010), 2222–2286. DOI: 10.1016/j.aim.2010.04.007
[12]P. T. Johnstone, Stone Spaces, Cambridge University Press, 1982. DOI: 10.1017/CBO9780511550850
[13]M. Piekosz, "Stone duality for Kolmogorov locally small spaces," Symmetry 13 (2021), no. 10, 1791. DOI: 10.3390/sym13101791
[14]H. A. Priestley, "Representation of distributive lattices by means of ordered Stone spaces," Bulletin of the London Mathematical Society 2 (1970), 186–190. DOI: 10.1112/blms/2.2.186
[15]M. Stone, "Topological representations of distributive lattices and Brouwerian logics," Transactions of the American Mathematical Society 41 (1937), 375–481. DOI: 10.1090/S0002-9947-1937-1501905-2
[16]R. Bandaru, R. Sirisetti, S. R. Kola, et al., "Stone paradistributive latticoids," European Journal of Pure and Applied Mathematics 18 (2025), no. 4. DOI: 10.29020/nybg.ejpam.v18i4.7135
[17]N. A. Bazhenov, I. Sh. Kalimullin, and M. V. Schwidefsky, "An effective version of the Stone duality," (2026). arXiv: 2604.04492
[18]Berger, "Stone duality for spectral sheaves and the patch monad," Journal of Pure and Applied Algebra 227 (2023), 107306. DOI: 10.1016/j.jpaa.2023.107306
[19]G. Bezhanishvili, L. Carai, and P. Morandi, "Duality theory for bounded lattices: A comparative study," (2025). arXiv: 2502.21307
[20]Craig, M. Haviar, and J. S. João, "Dual spaces of lattices and semidistributive lattices," (2025). arXiv: 2509.04417

International Journal of Mathematical Sciences and Computing (IJMSC)