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**THE PRINCETON overview will get effects. Get additional guidance for a great AP U.S. executive & Politics rating with 550 additional perform questions and answers.** This publication version has been optimized for electronic studying with cross-linked questions, solutions, and explanations.

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**Additional info for 550 AP U.S. Government & Politics Practice Questions**

**Example text**

50 (Nakayama’s Lemma). Let M be a nonzero finitely generated module over a ring A and I ⊆ A be an ideal such that aM = 0 for any a ∈ 1 + I. Then IM = M . Proof. Let M = Ae1 + . . + Aen . Assume that IM = M . Then ei = j αij ej with some αij ∈ I, or, equivalently, j (δij − αij )ej = 0. Let P be the n × n-matrix (δij − αij )ni,j=1 . Then P e = 0, where e is the column vector (e1 , . . , en ). 51. Let A be a ring, P = (aij ) ∈ Mat(n × n, A), and P = (Aji ) be the adjugate or classical adjoint matrix of P , where Aij = (−1)Mij is the (i, j) cofactor of P and Mij is the (i, j) minor of P , that is the determinant of the (n − 1) × (n − 1)-submatrix obtained from P by deleting row i and column j.

Un−1 ]. Let f (u1 , . . , un ) = aj uj11 . . ujnn , aj ∈ K \ {0} 38 IVAN ARZHANTSEV and let m1 , . . , mn−1 be positive integers. We define mn−1 1 u1 := u1 − um . n , . . , un−1 := un−1 − un i Let us substitute ui = ui + um n into f (u1 , . . , un ). Using vector notation, we put (m) = (m1 , . . , mn−1 , 1) and define (j) · (m) = j1 m1 + . . + jn−1 mn−1 + jn . Then we have f (u1 , . . , un ) = aj u(j)·(m) + g(u1 , . . , un−1 , un ) = 0, n where g(u1 , . . , un−1 , un ) is a polynomial in which no pure power of un appears.

Jn = kn . Proof. We have (j1 − k1 )d + (j2 − k2 )d2 + . . + (jn−1 − kn−1 )dn−1 + (jn − kn ) = 0. We may assume that jn−1 − kn−1 = 0. Then |(jn−1 − kn−1 )dn−1 | ≥ dn−1 and |(j1 −k1 )d+. +(jn−2 −kn−2 )dn−2 +(jn −kn )| ≤ (d−1)d+. +(d−1)dn−2 +(d−1) = = (d − 1)(1 + . . + dn−2 ) = dn−1 − 1, a contradiction. Let d be a positive integer greater than any component of any vector j with aj = 0 and let m = (d, d2 , . . , dn−1 , 1). Then (j) · (m) are pairwise distinct and we obtain an integral relation for un over K[u1 , .