# A First Course in Module Theory by Mike E Keating

By Mike E Keating

Some time past twenty years, there was nice development within the idea of nonlinear partial differential equations. This ebook describes the growth, concentrating on fascinating subject matters in fuel dynamics, fluid dynamics, elastodynamics and so on. It includes ten articles, each one of which discusses a really contemporary consequence bought through the writer. a few of these articles evaluate similar effects earrings 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 kinds for matrices; projective modules; tricks for the workouts

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**Extra resources for A First Course in Module Theory**

**Sample text**

1), and a has a trivial "factorization" with s = 0. (a) > 1. If a is already irreducible, it is its own one-term factorization. If a is reducible, a = be with neither b nor c a unit. (&) < ip(a) and 1 < ip(c) < <^(a), so, by induction hypothesis, both b and c already have factorizations into irreducible elements, which can be multiplied together to give a factorization of a. The uniqueness is established by induction on the number s of irre ducible factors. Suppose first that s = 1, so that a is irreducible.

Let V be a vector space V over a field F. , kt £ F so that v = kiXi + 1- ktxt. , fct £ F, then fcx = • ■ ■ = kt = 0. If V has a finite generating set X, then a finite basis of V can be obtained from the generating set by successively omitting elements. Moreover, any linearly independent subset Y of V can be extended to a basis by adding suitable members of X, and any two bases of V have the same number of members, this number being the dimension of V. Naturally enough, we refer to a finitely generated vector space as definite dimensional vector space.

On the other hand, we can interpret the product rs as the result of s acting on r and so turn R into a right R-module. R-module. Equally ubiquitous are the zero modules. For any ring R, the set {0} is both a left and a right R-module, with rO = 0 = Or always. We denote any zero module by 0 - thus we use the same notation for a zero module as we do for a zero element, but this should not cause any confusion in practice. Suppose that the ring of scalars R is commutative. Then we can convert any right module into a left module by the rule rm = mr for all r € R,m € M, and likewise, any left module is equally a right module.