A First Course in Module Theory by Mike E Keating

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By Mike E Keating

Long ago 20 years, there was nice development within the thought of nonlinear partial differential equations. This e-book describes the development, targeting fascinating issues in gasoline dynamics, fluid dynamics, elastodynamics and so forth. It comprises ten articles, each one of which discusses a really contemporary outcome bought via the writer. a few of these articles evaluation comparable 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; shows; diagonalizing and inverting matrices; becoming beliefs; the decomposition of modules; basic types for matrices; projective modules; tricks for the routines

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Then every ideal of R is principal. Proof It is obvious that the zero ideal 0 is principal, its unique generator being the zero element 0. Now suppose that 7 ^ 0 , and choose an element a of I so that 0 <

F[X]-module with X acting as A. Thus for m = Xm ( X } e M, = Again, any proper submodule L of M must be given by an eigenvector of A. The eigenvalues of A are the roots of det ( £ J =X2 + 1. X x Now suppose that F = R, the field of real numbers. Then A has no eigenvalues, so M cannot have any proper submodules, that is, it is a simple R[X]-module. This result is also intuitively true geometrically, since A corresponds to a rotation of the plane through it/2. However, if we take instead F = C, the complex numbers, then there are two eigenvalues, +i, —t, with respective eigenvectors v+ = ( \ I and V- = ( l J V -» We obtain two one-dimensional C-submodules of M, namely L + = Cv+ and L_ = Cv_.

If both are nonzero and b \ a, then the computation is again trivial: (a, b) = b. (r) < ip(b). Since a € Rb+Rr and r € Ra+Rb, we have Ra+Rb = Rb+Rr and so (o, 6) = (b,r). If r divides 6, we are done. If not, we write 6 = q\r+ri with

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