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. The notion of the approximation property is due to Michael Artin.

• Artin approximation theorem
• Popescu's theorem

• 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. Bibcode:1987InMat..88...39R. 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. Bibcode:1968InMat...5..277A. doi:10.1007/BF01389777. ISSN 0020-9910.