Al-Janabi, Hashmia Athrwi Ajaj (2026) Radical-Constrained Intersections of Direct Summands in Modules. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 7 (4). ISSN 2660-5309

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Abstract

We introduce the radical-constrained intersection property (RCI), a radical-localized weakening of the summand intersection property. A right R-module M is called RCI when, for every pair of direct summands A,B ≤⊕ M satisfying A ∩ B ≤ Rad(M), the intersection A ∩ B is a direct summand of Rad(M). This formulation localizes the splitting requirement itself inside the Jacobson radical and avoids lifting-type approximations by a possibly zero direct summand. We prove that every SIP-module is RCI, that the implication is strict, and that modules with semisimple or zero radical are RCI. The class is closed under direct summands, but not under arbitrary finite direct sums; a sufficient closure condition is obtained for Hom-orthogonal direct sums. We also give a quotient theorem under an explicit direct-summand lifting hypothesis and an idempotent characterization on the radical. The examples Z₂ ⊕ Z₄ and Z₄ ⊕ Z₈ show, respectively, that RCI need not imply SIP and that finitely generated modules need not be RCI. Relations with SISZ, SIPr, SIP-lifting, and radical-supplemented modules are discussed to clarify the position of the new condition in current decomposition theory.

Item Type:	Article
Uncontrolled Keywords:	Module, direct summand, Jacobson radical, RCI-module, SIP-module, SISZ-module, idempotent endomorphism, decomposition
Subjects:	H Social Sciences
Depositing User:	admin eprints
Date Deposited:	04 Nov 2026 09:21
Last Modified:	04 Nov 2026 09:21
URI:	http://eprints.umsida.ac.id/id/eprint/17461

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