Posted in Scientific Popular

New PDF release: Abstract Algebra (Course 311)

By Wilkins D.R.

Show description

Read Online or Download Abstract Algebra (Course 311) PDF

Best scientific-popular books

Download e-book for iPad: Annual Reports on NMR Spectroscopy, Vol. 23 by G. A. Webb

This publication is a part of a sequence on spectroscopy, and covers NMR reports of remoted spin-pairs within the strong nation, the oxidation country dependence of transition steel shieldings, the Cinderella nuclei, nuclear spin rest in natural structures, strategies of macromolecules and aggregates and the NMR of coals and coal items.

One World: The Interaction of Science and Theology by John C. Polkinghorne PDF

John C. Polkinghorne’s well known trilogy at the compatibility of faith and technological know-how is again in print. One global (originally released in 1986) introduces matters in technological know-how and faith that Dr. Polkinghorne to that end endured in technology and windfall and technology and construction. The books were broadly acclaimed separately and as a chain.

Additional info for Abstract Algebra (Course 311)

Sample text

Now r ∈ I, since r = j − qn, j ∈ I and qn ∈ I. But 0 ≤ r < n, and n is by definition the smallest strictly positive integer belonging to I. We conclude therefore that r = 0, and thus j = qn. This shows that I = nZ, as required. 3 Quotient Rings and Homomorphisms Let R be a ring and let I be an ideal of R. If we regard R as an Abelian group with respect to the operation of addition, then the ideal I is a (normal) subgroup of R, and we can therefore form a corresponding quotient group R/I whose elements are the cosets of I in R.

We say that an ideal I of the ring R is finitely generated if there exists a finite subset of I which generates the ideal I. 5 Let R be a unital commutative ring, and let X be a subset of R. Then the ideal generated by X coincides with the set of all elements of R that can be expressed as a finite sum of the form r1 x1 + r2 x2 + · · · + rk xk , where x1 , x2 , . . , xk ∈ X and r1 , r2 , . . , rk ∈ R. Proof Let I be the subset of R consisting of all these finite sums. If J is any ideal of R which contains the set X then J must contain each of these finite sums, and thus I ⊂ J.

Suppose that I +g = I. Now the only factors of f are constant polynomials and constant multiples of f , since f is irreducible. But no constant multiple of f can divide g, since g ∈ I. It follows that the only common factors of f and g are constant polynomials. Thus f and g are coprime. 12 that there exist polynomials h, k ∈ K[x] such that f h + gk = 1. But then (I +k)(I +g) = I +1 in K[x]/I, since f h ∈ I. Thus I +k is the multiplicative inverse of I + g in K[x]/I. We deduce that every non-zero element of K[x]/I is invertible, and thus K[x]/I is a field, as required.

Download PDF sample

Abstract Algebra (Course 311) by Wilkins D.R.

by Ronald

Rated 4.54 of 5 – based on 26 votes