By Mike E Keating

ISBN-10: 186094096X

ISBN-13: 9781860940965

Some time past 20 years, there was nice growth within the conception of nonlinear partial differential equations. This booklet describes the development, concentrating on attention-grabbing issues in gasoline dynamics, fluid dynamics, elastodynamics and so on. It comprises ten articles, every one of which discusses a truly contemporary consequence bought via the writer. a few of these articles evaluate similar effects jewelry and beliefs; Euclidean domain names; modules and submodules; homomorphisms; quotient modules and cyclic modules; direct sums of modules; torsion and the first decomposition; displays; diagonalizing and inverting matrices; becoming beliefs; the decomposition of modules; basic types for matrices; projective modules; tricks for the workouts

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Extra resources for A First Course in Module Theory

Example text

1 T h e o r e m Let F be a field, let A be an n x n matrix over F, and let M be the F[X)-module obtained from the vector space Fn with X acting as A. Then there is a bijective correspondence between (i) F[X)-submodules L of M, and 46 Chapter 3. Modules and Submodules (ii) F-subspaces U of Fn which are invariant under A, that is, AU C U. □ Now we have given the general description of submodules of modules defined by matrix actions, we look at some increasingly specific calculations. 10 Eigenspaces Given an n x n matrix A over a field F, an eigenspace for A is a nonzero subspace U of Fn with the property that there is a scalar A g F s o that Au = Aw for all u € U.

Proof =x By GCD 2, d € Ra+Rb and so Rd C Ra + Rb. For the reverse inclusion, suppose that r 6 Ra + Rb, with r = xa + yb for some x,y S R. By GCD 1, a — a'd and b = b'd for a',b' e R, and so r = (xa' + yb')d belongs to Rd. Chapter 2. Euclidean Domains 22

The notation (a, b) will be used for the standard choice of the greatest common divisor of a and b in the ring of integers or a polynomial ring, and for some arbitrary choice of a greatest common divisor in other rings, such as the Gaussian integers, where there is no evident "standard" choice. 7 Euclid's algorithm At this point, we should explain the origin of the term Euclidean domain. In Book VII, Propositions 1 and 2, of his Elements [Euclid], Euclid gives an algorithmic procedure for the computation of the greatest common divisor of a pair of integers.