A First Course in Module Theory by Mike E Keating

By Mike E Keating

Long ago 20 years, there was nice growth within the idea of nonlinear partial differential equations. This publication describes the growth, targeting fascinating themes in fuel dynamics, fluid dynamics, elastodynamics and so on. It comprises ten articles, each one of which discusses a truly fresh outcome acquired by way 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 varieties for matrices; projective modules; tricks for the workouts

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Extra info for A First Course in Module Theory

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It actually is irreducible, since neither element of Z2 is a root. 11 Residue rings of Euclidean domains The fact that every ideal of a Euclidean domain is principal, together with the Unique Factorization Theorem, leads to a good description of the residue rings of a Euclidean domain. We will postpone the general treatment of this topic until we discuss cyclic modules in Chapter 6, since the discussion can be simplified when we have some machinery from mod­ ule theory at our disposal. In the remainder of this chapter, we will show how fields arise as residue rings of Euclidean domains and we will give an explicit description of the residue rings of polynomial rings.

As our main interest is with modules over commutative rings, for which we use the lefthanded notation, we will as a rule give only the left-handed versions of such statements and definitions. The reader should have no problem in providing the right-handed versions where desired. 6. Sum and intersection 41 When the ring of scalars R is commutative, then any left module M can be regarded as a right module, and any submodule of M as a left module is equally a submodule of M as a right module. A left module M is always a submodule of itself.

Sometimes it is convenient to allow non-unital modules, but we shall not do so in this text. The ring R is called the ring of scalars for M. If the elements of R act on the right of M, then we obtain a right module, and the rules for scalar multiplication are changed accordingly: SMR 1: m(rs) = (mr)s for all m e M and r,s 6 R. SMR 2: (m + n)r = mr + nr and m(r + s) = mr + ms for all m,n e M and all r,s € R. SMR 3: m l = m for all m in M, where 1 is the identity element in R. Extremely important examples of R-modules arise from the ring R itself, which can be regarded either as a left module or as a right module.

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