By Douglas R. Farenick (auth.)

The objective of this e-book is twofold: (i) to provide an exposition of the elemental concept of finite-dimensional algebras at a levelthat isappropriate for senior undergraduate and first-year graduate scholars, and (ii) to supply the mathematical beginning had to organize the reader for the complicated learn of an individual of a number of fields of arithmetic. the topic below examine is certainly not new-indeed it's classical but a e-book that gives a simple and urban therapy of this idea turns out justified for a number of purposes. First, algebras and linear trans formations in a single guise or one other are regular positive aspects of assorted components of contemporary arithmetic. those comprise well-entrenched fields akin to repre sentation idea, in addition to more recent ones similar to quantum teams. moment, a research ofthe effortless concept offinite-dimensional algebras is very important in motivating and casting gentle upon extra refined issues resembling module conception and operator algebras. certainly, the reader who acquires a great figuring out of the elemental idea of algebras is wellpositioned to ap preciate leads to operator algebras, illustration idea, and ring concept. In go back for his or her efforts, readers are rewarded through the implications themselves, numerous of that are primary theorems of outstanding elegance.

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IF [x ] is a principal ideal domain; that is, if J is a proper ideal of IF [x], then there exists a polynomial f E IF [z] such that J is the principal ideal (f) = {h(x )f(x ) : hE IF [xl}. 2. If a polynomial f E IF [x], is irreducible, then the quotient ring IF [xli (f) is a field extension of IF. Recall that the Euclidean division algorithm in IF [xl asserts that if f is a fixed polynomial , say n f( x) = L aj x j , j=O then for every g E IF [x] there exist unique polynomials q (th e quotient) and r (the remainder) such that (i) g(x) = f( x)q(x) + r( x) , and (ii) r(x) = 0 or degr < degf , where deg f denot es the degree of the polynomial f.

We are using the same notat ion to denote th e element 1 of the field IF and the element 1 of the algebra ~. ) If an algebra has a multiplicative ident ity, then it is easy to verify t hat t his identity is necessarily unique. An algebra having a multiplicative identity is said to be unital. Algebras are much like number syste ms: they are struct ures in which addition, subtract ion, and multi plication (associative but possibly noncommutative) are valid operations. Th e operation of division in algebras does not really exist in genera l; at best , one is interested in whet her or not a part icular element might be inverti ble.

More special still is a linear order, which is a partial order that satisfies one more property: (iv) (comparability) for all a, s « 6, either a j b or b j a. To get a feel for the definitions, the motivating examples one has in mind are: (1) the power set P(Y) of all subsets of a nonempty set Y, where A j B , for subsets A and B of Y, is taken to mean A <:;;; B; and (2) the real numbers JR, where x j y for x, y E JR means "less than or equal to" (x :::; y). In the first case the relation j is a partial order, but not a linear order; in the second case, though, j is a linear order.