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Tangential idealizers and differential ideals. (English) Zbl 1342.13039
Let $$k\subset R$$ be an extension of commutative Noetherian rings with identity. If $$I$$ is an ideal of $$R$$ and $$N$$ a submodule of an $$R$$-module $$M$$, then the tangential idealizer (over $$k$$) of $$I$$ with respect to $$N$$ is defined as $$T_{R|k}^{M}(I, N)=\{d\in\mathrm{Der}_k(R, M)\mid d(I)\subseteq N\}$$. If $$M=R$$ and $$N=I$$, one writes $$T_{R|k}(I)$$ for $$T_{R|k}^{M}(I, N)$$.
The paper under review establishes some properties of tangential idealizers; in particular, it is shown that if $$I=\bigcap_{i=1}^tQ_i$$ is a primary decomposition of an ideal $$I$$ without embedded components, then $$T_{R|k}(I)=\bigcap_{i=1}^{t}T_{R|k}(Q_{i})$$. Moreover, if $$I$$ is non-differential (i. e., it is not invariant under every $$k$$-derivation of $$R$$), then the last intersection taken over $$\{i\mid Q_i$$ is a non-differential$$\}$$ is a minimal primary decomposition of $$T_{R|k}(I)$$ in $$\mathrm{Der}_k(R, R)$$. The author also describes ideals $$I$$ whose radicals, as well as ordinary and symbolic powers, have the same tangential idealizer $$T_{R|k}(I)$$. The last part of the paper deals with differential ideals (that is, ideals $$I$$ such that $$d(I)\subseteq I$$ for every $$d\in\mathrm{Der}_k(R, R)$$). Using properties of tangential idealizers, the author gives a characterization of differential ideals and proves that if $$I$$ is an ideal of $$R$$ having no embedded primary components, then $$I$$ is differential if and only if all its primary components are differential.

##### MSC:
 13N15 Derivations and commutative rings 13C05 Structure, classification theorems for modules and ideals in commutative rings 13C13 Other special types of modules and ideals in commutative rings 13N99 Differential algebra
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##### References:
 [1] Brumatti, P; Lequain, Y, Maximally differential ideals, J. Algebra, 168, 172-181, (1994) · Zbl 0840.13014 [2] Brumatti, P; Simis, A, The module of derivations of a Stanley-Reisner ring, Proc. Am. Math. Soc., 123, 1309-1318, (1995) · Zbl 0826.13006 [3] Camacho, C; Sad, P, Invariant varieties through singularities of holomorphic vector fields, Ann. Math., 115, 579-595, (1982) · Zbl 0503.32007 [4] Corso, A., Huneke, C., Katz, D., Vasconcelos, W.V.: Integral closure of ideals and annihilators of homology. In: Commutative Algebra, Lect. Notes Pure Appl. Math., Chapman and Hall/CRC, Boca Raton, FL, 244, 33-48 (2006) · Zbl 1119.13006 [5] Eisenbud, D.: Commutative algebra with a view toward algebraic geometry, graduate texts in mathematics, vol. 150. Springer, Berlin (1995) · Zbl 0819.13001 [6] Hart, R.: Derivations on commutative rings. J. London Math. Soc. (2) 8, 171-175 (1974) · Zbl 0282.13002 [7] Hauser, H; Risler, J-J, Dérivations et idéaux réels invariants, Publ. Res. Inst. Math. Sci., 35, 585-597, (1999) · Zbl 0984.32004 [8] Hochster, M, Criteria for equality of ordinary and symbolic powers of primes, Math. Z., 133, 53-65, (1973) · Zbl 0251.13012 [9] Huneke, C., Swanson, I.: Integral closure of ideals, rings, and modules. London Math. Soc. Lecture Note Ser., vol. 336, Cambridge Univ. Press, Cambridge (2006) · Zbl 1117.13001 [10] Johnson, J, A notion on Krull dimension for differential rings, Comment. Math. Helv., 44, 207-216, (1969) · Zbl 0179.34401 [11] Kaplansky, I.: An introduction to differential algebra. Hermann, Paris (1957) · Zbl 0083.03301 [12] Keigher, W.: Prime differential ideals in differential rings. In: Contributions to Algebra: a Collection of Papers Dedicated to Ellis Kolchin (Bass, Cassidy and Kovacic, eds.) Academic Press, New York (1977) · Zbl 0369.12015 [13] Lequain, Y, Differential simplicity and complete integral closure, Pacific J. Math., 36, 741-751, (1971) · Zbl 0188.09702 [14] Looijenga, E.J.N.: Isolated singular points on complete intersections. London Math. Soc. Lecture Note Ser. vol. 77. Cambridge Univ. Press, Cambridge (1984) · Zbl 0552.14002 [15] Maloo, AK, Maximally differential ideals, J. Algebra, 176, 806-823, (1995) · Zbl 0969.13009 [16] Miranda-Neto, CB, Vector fields and a family of linear type modules related to free divisors, J. Pure Appl. Algebra, 215, 2652-2659, (2011) · Zbl 1229.13021 [17] Patil, DP; Singh, B, Remarks on maximally differential prime ideals, J. Algebra, 83, 387-392, (1983) · Zbl 0514.13020 [18] Raudenbush, HW, Ideal theory and algebraic differential equations, Trans. Am. Math. Soc., 36, 361-368, (1934) · Zbl 0009.10004 [19] Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 27, 265-291 (1980) · Zbl 0496.32007 [20] Seibt, P, Differential criteria on the permissibility of a blowing-up, J. Algebra, 66, 484-491, (1980) · Zbl 0469.13002 [21] Seidenberg, A, Derivations and integral closure, Pacific J. Math., 16, 167-173, (1966) · Zbl 0133.29202 [22] Seidenberg, A, Differential ideals in rings of finitely generated type, Am. J. Math., 89, 22-42, (1967) · Zbl 0152.02905 [23] Siebert, Th, Lie algebras of derivations and affine algebraic geometry over fields of characteristic 0, Math. Ann., 305, 271-286, (1996) · Zbl 0858.17018 [24] Simis, A.: Differential idealizers and algebraic free divisors. In: Commutative Algebra, Lect. Notes Pure Appl. Math., 244, Chapman and Hall/CRC, Boca Raton, FL, pp. 211-226 (2006) · Zbl 1099.13030 [25] Terao, H, Arrangements of hyperplanes and their freeness I, II, J. Fac. Sci. Univ. Tokyo Sect. Math., 27, 293-320, (1980) · Zbl 0509.14006 [26] Vasconcelos, WV, Derivations of commutative Noetherian rings, Math. Z., 112, 229-233, (1969) · Zbl 0181.05201 [27] Vasconcelos, W.V.: Integral closure. Rees algebras, multiplicities, algorithms, Springer monographs in mathematics. Springer, New York (2005) · Zbl 1082.13006
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