The connection between harmonic functions and different types of ideals in commutative algebra

This paper investigates the relationship between harmonic functions and several classes of ideals in polynomial rings. Focusing on prime, primary, maximal, and weakly primary ideals (see [M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra (Addison-Wesley, 1969); D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry (Springer-Verlag, New York, 1995); O. Zariski and P. Samuel, Commutative Algebra, Vol. II (Van Nostrand, Princeton,1960)]), we analyze how these algebraic structures reflect and influence the behavior of harmonic functions. Through theoretical analysis and illustrative examples, we examine the algebraic foundations of harmonics within the context of commutative algebra. Furthermore, we discuss the role of D-modules as a powerful tool for studying differential operators, particularly the Laplacian, in relation to harmonic functions (see [J.-E. Bj[Formula: see text]rk, Rings of Differential Operators, North-Holland Mathematical Library, Vol. 21 (North Holland, Amsterdam, 1979); S. C. Coutinho, A Primer of Algebraic D-Modules (Cambridge University Press, 1995); H. Johannes, P. Elizabeth, S. Anna-Laura and Z. Simone, D-modules and analytic representation of differential equations, preprint (2023); V. A. Lunts and A. Rosenberg, Differential operators and D-module structures in noncommutative geometry, Adv. Math. 415 (2023) 108901; M. Saito, K. Takeuchi and H. Terao, D-modules and Microlocal Geometry, Springer Monographs in Mathematics (Springer, 2022)]).