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(*
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Ideals for commutative rings
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$Id$
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Author: Clemens Ballarin, started 24 September 1999
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*)
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Ideal = Ring +
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consts
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ideal_of :: ('a::ringS) set => 'a set
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is_ideal :: ('a::ringS) set => bool
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is_pideal :: ('a::ringS) set => bool
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defs
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is_ideal_def "is_ideal I == (ALL a: I. ALL b: I. a + b : I) &
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11093
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(ALL a: I. - a : I) & 0 : I &
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7998
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(ALL a: I. ALL r. a * r : I)"
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ideal_of_def "ideal_of S == Inter {I. is_ideal I & S <= I}"
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is_pideal_def "is_pideal I == (EX a. I = ideal_of {a})"
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(* Principle ideal domains *)
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axclass
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pid < domain
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pid_ax "is_ideal I ==> is_pideal I"
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end |