By Mike E Keating
Long ago 20 years, there was nice development within the idea of nonlinear partial differential equations. This e-book describes the development, targeting fascinating subject matters in gasoline dynamics, fluid dynamics, elastodynamics and so forth. It comprises ten articles, every one of which discusses a truly contemporary consequence bought by means of the writer. a few of these articles evaluation comparable 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; common types for matrices; projective modules; tricks for the workouts
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Additional info for A First Course in Module Theory
If m < n then we can take q = 0 and r = f. This remark also covers the initial case in the induction, when m = 0, for if m = n = 0 then both / and g are units in F[X] and the assertion is trivial. Suppose now that m > n, and put / = / - fng^n1Xm~ng. By con struction, the coefficient of Xm in / is 0, so deg(/) < m. By induction hypothesis, / = qg + r with deg(r) < deg(g) (or r = 0), and a rearrange ment gives the desired form for / . Unlike the situation for Z, the quotient and remainder are unique in polynomial rings.
Thus, in the case of main interest in this text, we need not distinguish between left and right modules. We will use the left-handed notation for a module over a commutative ring. 10 below, and from Chapter 14. The modules over one special kind of ring are familiar from elementary linear algebra. When F is a field, an F-module is the same thing as a vector space over F . 1), the theory of modules over Euclidean domains includes the theory of vector spaces. However, we need to assume a prior knowledge of vector space theory so that we can introduce some of our basic examples of modules.
Modules and Submodules 40 case, the corresponding F[X]-module is called the trivial F[X]-module, on which X acts as 0. Be careful not to confuse the trivial module with the zero module; the latter has only one element 0. When n = 1, the vector space F1 is simply the field F regarded as a vector space over itself. ) A 1 x 1 matrix is, for all practical purposes, the same thing as an element A in F, so we find that for each choice of A, F can be made into an F[Af]-module with X acting as A. The action of a polynomial / is given explicitly by fk = f0k + fx\k + ■■■ + fiXk + ■■■ + fnXnk, keF.
A First Course in Module Theory by Mike E Keating