Multilinear algebra by Northcott D.G.

By Northcott D.G.

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Thus if I1 ⊆ I2 ⊆ I3 ⊆ . . is a chain of ideals in R, then there is some m for which Ik = Im for all k m. Note: Commutative rings satisfying the ACC are called Noetherian. To understand what the ACC means it may be helpful to look at an example of a ring in which it does not hold. 2 Let C(R) denote the ring of continuous functions from R to R with addition and multiplication defined by (f + g)(x) = f(x) + g(x); (fg)(x) = f(x)g(x), for f, g ∈ C(R), x ∈ R. For n = 1, 2, 3, . . , define In to be the subset of C(R) consisting of those functions that map every element of the interval − n1 , n1 to 0.

The irreducible elements of Z are p and −p, for p prime. 2. In Q[x], the units are the non-zero constant polynomials. The associates of a non-zero non-constant polynomial f(x) are the polynomials of the form af(x) where a ∈ Q× . So x2 + 2 is associate to 3x2 + 6, 21 x2 + 1, etc. 3. In Z the irreducible elements are the integers p and −p where p is a prime numbers. The prime elements of Z are exactly the irreducible elements - the prime numbers and their negatives. 2 An integral domain R is a unique factorization domain if the following conditions hold for each element a of R that is neither zero nor a unit.

Pr (x) and f(x) = q1 (x)q2 (x) . . qs (x) are two such expressions, with s r. Then q1 (x)q2 (x) . . qs (x) belongs to the ideal p1 (x) of F[x]. Since this ideal is prime (as p1 (x) is irreducible) this means that either q1 (x) ∈ p1 (x) or q2 (x) . . qs (x) ∈ p1 (x) . Repeating this step leads to the conclusion that at least one of the qi (x) belongs to p1 (x) . After reordering the qi (x) if necessary we have q1 (x) ∈ p1 (x) . Since q1 (x) is irreducible this means q1 (x) = u1 p1 (x) for some unit u1 .

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