Commutative Rings |
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Page 83
... prime ideal contains a minimal prime ideal . 4. ( a ) Under the hypotheses of Theorem 110 , prove that any non- zero element of R lies in only a finite number of minimal prime ideals . ( Hint : review the proof of Theorem 110. ) ( b ) ...
... prime ideal contains a minimal prime ideal . 4. ( a ) Under the hypotheses of Theorem 110 , prove that any non- zero element of R lies in only a finite number of minimal prime ideals . ( Hint : review the proof of Theorem 110. ) ( b ) ...
Page 119
... minimal prime ideal ( namely 0 ) . This can be formulated in R itself as the state- ment that each M contains a unique minimal prime ideal , and we use this as the hypothesis of the next theorem . Theorem 167. Let R be a Noetherian ring ...
... minimal prime ideal ( namely 0 ) . This can be formulated in R itself as the state- ment that each M contains a unique minimal prime ideal , and we use this as the hypothesis of the next theorem . Theorem 167. Let R be a Noetherian ring ...
Page 132
... ideal M ; ( 2 ) Every minimal prime ideal in R is finitely generated ; ( 3 ) Every invertible ideal in R is principal . Proof . The necessity needs little attention . Every localization of a UFD is a UFD ( Ex . 3 in §1-4 ) ; in a UFD every ...
... ideal M ; ( 2 ) Every minimal prime ideal in R is finitely generated ; ( 3 ) Every invertible ideal in R is principal . Proof . The necessity needs little attention . Every localization of a UFD is a UFD ( Ex . 3 in §1-4 ) ; in a UFD every ...
Contents
Preface | 1 |
Noetherian Rings | 47 |
Macaulay Rings and Regular Rings | 84 |
Copyright | |
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A/xA a₁ algebraic annihilator ascending chain condition assume Bézout coefficients commutative ring contradiction Dedekind deduce dim(R disjoint domain with quotient equation exact sequence FFR module finite number finitely generated non-zero finitely generated R-module G-domain G-ideal Gorenstein Hence Hilbert ring Hint hypothesis id(A implies induction integral domain integrally closed intersection invertible ideal isomorphic Jacobson radical Krull Let RCT Macaulay ring maximal ideal maximal R-sequence maximal with respect minimal prime ideal multiplicatively closed set Nakayama lemma Noetherian domain Noetherian ring non-zero element non-zero ideal non-zero prime ideal number of prime one-dimensional P₁ polynomial ring prime ideal containing principal ideal theorem principal prime proof of Theorem properly contained Prove Prüfer Q₁ quotient field radical ideals rank(P regular local ring Remark ring with maximal satisfies the ascending submodule Suppose Theorem 152 Theorem 81 V₁ valuation domain write x₁ zero-divisor