# Download A First Course in Module Theory by Mike E Keating PDF

By Mike E Keating

Long ago twenty years, there was nice development within the concept of nonlinear partial differential equations. This booklet describes the development, concentrating on attention-grabbing subject matters in fuel dynamics, fluid dynamics, elastodynamics and so on. It comprises ten articles, each one of which discusses a truly contemporary consequence bought by means of 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; common varieties for matrices; projective modules; tricks for the routines

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Show that if £ is a nonzero element of the additive group Z p , then x is a generator of Z p as a Z-module. Deduce that Z p is a simple Z-module. 2. Remark this result is more frequently given in the form "a cyclic group of prime order has no nontrivial subgroups". 3 49 Let A be an abelian group and let r 6 Z. } is a subgroup of A. Take A - Z m - Prove that rA = A <=> (r, m) = 1 and that rA = 0 <=> m divides r. R-module. 7 is a submodule of M. Suppose that M = R/Rx for some x e R. Determine conditions on r so that (i) rM = M and (ii) rM = 0.

2. 2 37 Additive groups Next, we show how an additive group can be viewed as a Z-module. 10), this observation will allow us to obtain results about abelian groups from our general theory of modules over Euclidean domains. 1). Intuitively, the action of a positive integer n on an element a in A is given by na = a + ■■■ + a, where there are n a's in the sum. A more formal inductive definition runs as follows. To start, define 0a = 0, where the first "0" is the zero in Z and the second "0" is the zero in A.

Deduce that f{X) is irreducible if and only if f(Y) is irreducible. Let p € E be a prime number. Prove that the polynomial Xp~ * + • • ■ + X + 1 = (X" - 1)/(X - 1) is irreducible in Q[X]. Show that the polynomial ring F[X, Y] in two variables over a field is not a Euclidean domain. Let R = Z[%/—5]. Show that 3, 2+\J—5 and 2 — yj— 5 are all irreducible elements of R and that no two of them are associates. Verify that 3 2 = (2 + v / ^ 5 ) ( 2 - V ^ 5 ) . This means that unique factorization does not hold in R and hence that R cannot be a Euclidean domain.