By Mike E Keating
Some time past twenty years, there was nice development within the conception of nonlinear partial differential equations. This e-book describes the development, concentrating on attention-grabbing themes in fuel dynamics, fluid dynamics, elastodynamics and so on. It comprises ten articles, each one of which discusses a truly contemporary end result received via the writer. a few of these articles overview 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 varieties for matrices; projective modules; tricks for the routines
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Additional resources for A First Course in Module Theory
Chapter 4 Homomorphisms We next introduce a fundamental concept in module theory, that of a homomorphism, which is a map from one module to another that respects the addition and scalar multiplication. A knowledge of the homomorphisms be tween two modules allows us to compare their internal structures. In later chapters, we use an analysis of homomorphisms to obtain the fundamental results on the structure of a module over a Euclidean domain. Just as a vector space over a field F is a special type of module, a linear transformation between vector spaces is another name for a homomorphism between them.
Thus a Z-module ho momorphism is another name for a group homomorphism from M to N. ) Here are three homomorphisms that are always present. • Given any module M over any ring R, the identity homomorphism idM : M ->■ M is defined by idM (TTI) = m for all m € M. • Given a submodule L of M, there is an inclusion homomorphism inc : L ->■ M, defined by inc(l) = I for all / e L. At first sight, it may seem pointless to give names to these "do nothing" maps, but there are circumstances where it is very useful to be able to distinguish between an element of L regarded as an element of L and the same element regarded as an element of M.
Z-modules. 2 above, a Z-module A is the same thing as an abelian group, written additively. A subgroup B of A, is, by definition, a subset of A which satisfies conditions SubM 1 and 2. However, if B does satisfy these conditions, then it also satisfies SubM 3, since scalar multiplication by an integer is essentially repeated addition or subtraction. Thus, a Z-submodule of A is the same thing as a subgroup of A • Vector spaces. Let F be a field. Then an F-module is a vector space V over F, and an F-submodule W of V is more familiarly called a subspace of V.
A First Course in Module Theory by Mike E Keating