Akademik Veri Yönetim Sistemi
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, sa.3, 2025 (SCI-Expanded)
In this paper, we study some variations of perspectivity of modules. We investigate the relationships between variations of perspectivity of a module and almost uniqueness of direct complements of a module. We show that there exist s-perspective modules, and so almost dual perspective modules, whose direct complements are not almost unique. Also we construct almost dual perspective and D3-modules, namely s-perspective modules whose direct complements are not almost unique. Moreover, we examine the direct sums of s-perspective modules and the modules whose direct complements are almost unique by using pairwise orthogonal primitive idempotents and m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{m}$$\end{document}-local components of any module M over a commutative ring. Finally, we prove some structural results over commutative Dedekind domains. Let R be a commutative Dedekind domain with quotient field Q. Let M be a non-zero injective R-module. Then direct complements of M are almost unique if and only if either M is a torsion module such that every non-zero P-primary component is isomorphic to R(P infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(P<^>\infty )$$\end{document} or M congruent to Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\cong Q$$\end{document}. As a consequence we obtain that direct complements of M are almost unique if and only if for every non-zero prime ideal P, there exist a is an element of{0,1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in \{0, 1\}$$\end{document} and a non-negative integer n such that TP(M)congruent to(R(P infinity))a circle plus(R/PnR)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_P(M)\cong (R(P<^>\infty ))<^>a\oplus (R/P<^>nR)$$\end{document} if and only if M is a D3-module, where R is a commutative Dedekind domain and M is a non-zero torsion R-module. Let R be a discrete valuation ring with maximal ideal m=pR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{m}=pR$$\end{document}. Let M be a non-zero reduced R-module which is not torsion-free. We show that direct complements of M are almost unique if and only if M congruent to R/mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\cong R/\textrm{m}<^>n$$\end{document} for some positive integer n.