Approximation property (ring theory)

In algebra, a commutative Noetherian ring A is said to have the approximation property with respect to an ideal I if each finite system of polynomial equations with coefficients in A has a solution in A if and only if it has a solution in the I-adic completion of A.[1][2] The notion of the approximation property is due to Michael Artin.

This article is about the concept in ring theory. For the concept in functional analysis, see approximation property.

See also

• Artin approximation theorem
• Popescu's theorem

Notes

1. Rotthaus, Christel (1997). "Excellent Rings, Henselian Rings, and the Approximation Property". Rocky Mountain Journal of Mathematics. 27 (1): 317–334. doi:10.1216/rmjm/1181071964. JSTOR 44238106.
2. "Tag 07BW: Smoothing Ring Maps". The Stacks Project. Columbia University, Department of Mathematics. Retrieved 2018-02-19.

References

• Popescu, Dorin (1986). "General Néron desingularization and approximation". Nagoya Mathematical Journal. 104: 85–115. doi:10.1017/S0027763000022698.
• Rotthaus, Christel (1987). "On the approximation property of excellent rings". Inventiones Mathematicae. 88: 39–63. doi:10.1007/BF01405090.
• Artin, M (1969). "Algebraic approximation of structures over complete local rings". Publications Mathématiques de l'IHÉS. 36: 23–58. doi:10.1007/BF02684596. ISSN 0073-8301.
• Artin, M (1968). "On the solutions of analytic equations". Inventiones Mathematicae. 5 (4): 277–291. doi:10.1007/BF01389777. ISSN 0020-9910.