Abstract
By a ring we mean a commutative ring with identity. A ring is said to be normal if it is integrally closed in its total quotient ring. By a domain we mean an integral domain. By a prime ideal (resp: a maximal ideal) in a ring A we mean an ideal P in A such that A/P is a domain (resp: a field); note that then P ≠ A. For any ideal P in a ring A, by rad A P or rad P we denote the radical of P in A. Let A be a ring and let P be an A-module; for any subset Q of P, by QA we denote the A-submodule of P generated by Q; for any elements x1 , ..., x n in P, by (x1 , ..., x n )A we denote the A-submodule of P generated by x1 , ..., x n ; elements x1 , ..., x n in P are said to form an A-basis (or simply, a basis) of P if P = (x1 , ..., x n )A; P is said to be a finite A-module if P has a finite A-basis. For any subset P of a ring B and any element x in B, by xP or Px we denote the subset {xy: y ∈ P} of B; note that if P is an A-submodule of B for a subring A of B then xP is an A-submodule of B, and if moreover (x1 , ..., x n ) is an A-basis of P then (xx1 , ..., xx n ) is an A-basis of xP. Given a ring A, let N be the set of all nonnegative integers n such that there exists a chain of distinct prime ideals P0 ⊂ P1 ⊂ ⋯ ⊂ P n in A; we define: dim A = − ∞ if N = ø, dim A = the greatest integer in N if N is a nonempty finite set, and dim A = ∞ if N is an infinite set.
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© 1998 Springer-Verlag Berlin Heidelberg
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Abhyankar, S.S. (1998). Local Theory. In: Resolution of Singularities of Embedded Algebraic Surfaces. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03580-1_2
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DOI: https://doi.org/10.1007/978-3-662-03580-1_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08351-8
Online ISBN: 978-3-662-03580-1
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