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On Zariski pseudo-primary radical of elements of lattice modules

Pradip Girase1iD and Narayan Phadatare2iD

  1. 1 Department of Mathematics, K. K. M. College, Manwath, Dist. Parbhani (M.S.) – 431505, India.
  2. 2 Department of Mathematics, SKES's Govindram Seksaria Science College (Autonomous), Belagavi, Karnataka, India.
Published 31 Aug 2026Asia Mathematika, volume 10, issue 2, pages 72–82 (2026)doi:10.5281/zenodo.23122374
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Abstract

Let L be a C-lattice and M be a lattice module over L. For an element N ∈ M, the meet of all pseudo-primary elements P of M satisfying √(N : 1M) ≤ √(P : 1M) is called the Zariski pseudo-primary radical of N, denoted by zps√N. This paper introduces the notion of the Zariski pseudo-primary radical of elements in lattice modules and presents a systematic study of its algebraic properties. Several characterizations and basic results are established, highlighting its connections with pseudo-primary elements, pseudo-primary radicals and the pseudo-primary spectrum of lattice modules. Also, we study lattice modules whose pseudo-primary spectrums endowed with the Zariski-like topology are Noetherian spaces.

Lattice modulesPseudo-primary elementsPseudo-primary spectrumZariski pseudo-primary radical

1. Introduction

The notion of primary-like submodules was introduced and studied by H. F. Moghimi and F. Rashedi in [17]. Subsequently, in [19], the authors developed the Zariski topology on the primary-like spectrum of a module and investigated several of its topological properties. Later, they introduced in [18] the concept of the Z-radical of submodules and studied the primary-like spectrum from the viewpoint of Noetherian spaces. Motivated by these developments, the concept of pseudo-primary elements was introduced for lattice modules by A. V. Bingi et al. in [6], extending the theory of primary-like submodules to the framework of lattice modules. Furthermore, P. B. Girase et al. studied the Zariski prime radical elements of lattice modules in [12], while V. Borkar et al. investigated the Zariski second radical elements and associated properties in [7].

This page shows an excerpt of the introduction. The full text, including the theorems, proofs and examples, is available in the PDF.

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Cite this article

Girase, P. & Phadatare, N. (2026). On Zariski pseudo-primary radical of elements of lattice modules. Asia Mathematika, 10(2), 72–82. https://doi.org/10.5281/zenodo.23122374

Publication history

Received21 Jul 2026
Accepted02 Aug 2026
Published31 Aug 2026