# Algebraists' Homage: Papers in Ring Theory and Related by S. A. Amitsur, D. J. Saltman, George B. Seligman

By S. A. Amitsur, D. J. Saltman, George B. Seligman

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Qn of S such that Qi ∩ R = Pi for all 1 ≤ i ≤ n. Proof: By induction on n it suffices to prove the case n = 2. By localizing both R and S at the multiplicatively closed set R \ P2 , without loss of generality R is a local integrally closed domain with maximal ideal P2 . It suffices to prove that P1 contracts from a prime ideal contained in Q2 , or that P1 contracts from a prime ideal in the ring extension SQ2 . 4 it suffices to prove that P1 SQ2 ∩ R = P1 . 3. Lying-Over, Incomparability, Going-Up, Going-Down 33 Let r ∈ P1 SQ2 ∩ R.

Without loss of generality deg(e1 ) > · · · > deg(en ). If deg(e1 ) > 0, the homogeneous part of e2 − e of degree 2 deg(e1 ) is exactly e21 , and since e2 − e = 0 and R is reduced, we get a contradiction. Similarly if deg(en ) < 0 we get a contradiction. This forces n = 1 and e to be homogeneous of degree 0, whence proving (3). Assume (3). To prove (4), by symmetry it suffices to prove (4) for i = 1. Consider the idempotent e = (1, 0, . . 13). By assumption e is homogeneous. 3. Integral closure and grading 37 elements a, b ∈ R, with b a non-zerodivisor in R such that e = a/b.

If (F1 , . . , Fm ) = R1 R, then (F1 , . . , Fm ) = R1 R. Proof: By assumption there exists n such that R1n ⊆ (F1 , . . , Fm ). By homogeneity, R1n R ⊆ R1n−1 (F1 , . . , Fm ). This proves that (F1 , . . 5 that R1 R ⊆ (F1 , . . , Fm ). But R1 R is a radical ideal, so R1 R ⊆ (F1 , . . , Fm ) ⊆ (F1 , . . , Fm ) ⊆ R1 R, which finishes the proof. , µ(m) > dim R, and let I be an ideal of finite projective dimension. Then m(I : m) = mI and I : m is integral over I. Proof: Without loss of generality I = R.