Abstract
Let \(R\) be a commutative Noetherian ring, \(I\) an ideal of \(R\), and \(M\) an \(R\)-module. The ambiguous structure of \(I\)-transform functor \(D_{I}(-)\) makes the study of its properties attractive. In this paper we gather conditions under which \(D_{I}(R)\) and \(D_{I}(M)\) appear in certain roles. It is shown under these conditions that \(D_{I}(R)\) is a Cohen–Macaulay ring, regular ring, \(\cdots\) and \(D_{I}(M)\) can be regarded as a Noetherian, flat, \(\cdots R\)-module. We also present a primary decomposition of zero submodule of \(D_{I}(M)\) and secondary representation of \(D_{I}(M)\).
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REFERENCES
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra (Addison-Wesley, 1969).
K. Bahmanpour, ‘‘Exactness of ideal transforms and annihilators of top local cohomology modules,’’ J. Korean Math. Soc. 52, 1253–1270 (2015). https://doi.org/10.4134/JKMS.2015.52.6.1253
M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, Vol. 60 (Cambridge Univ. Press, Cambridge, 1998).
W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, Vol. 39 (Cambridge Univ. Press, New York, 1998).
L. Z. Chu, V. H. Jorge Pérez, and P. H. Lima, ‘‘Ideal transforms and local cohomology defined by a pair of ideals,’’ J. Algebra Its Appl. 17, 1850200 (2015). https://doi.org/10.1142/S0219498818502006
A. Grothendieck, Local Cohomology, Lecture Notes in Mathematics, Vol. 41 (Springer, Berlin, 1967).
R. Hartshorne, ‘‘Cohomological dimension of algebraic varieties,’’ Ann. Math. 88, 403–450 (1968). https://doi.org/10.2307/1970720
H. Matsumura, Commutative Ring Theory (Cambridge Univ. Press, Cambridge, 1986).
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MSC2010 numbers: 13D45; 14B15; 13E05
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Sadegh, Y., A’zami, J. & Yazdani, S. On the Ideal Transforms Defined by an Ideal. J. Contemp. Mathemat. Anal. 57, 399–404 (2022). https://doi.org/10.3103/S1068362322060073
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DOI: https://doi.org/10.3103/S1068362322060073