By Matej Brešar
Delivering an effortless advent to noncommutative jewelry and algebras, this textbook starts with the classical thought of finite dimensional algebras. simply after this, modules, vector areas over department earrings, and tensor items are brought and studied. this is often by way of Jacobson's constitution thought of earrings. the ultimate chapters deal with loose algebras, polynomial identities, and jewelry of quotients.
Many of the consequences aren't offered of their complete generality. really, the emphasis is on readability of exposition and straightforwardness of the proofs, with numerous being assorted from these in different texts at the topic. necessities are saved to a minimal, and new options are brought progressively and are rigorously inspired. creation to Noncommutative Algebra is hence obtainable to a large mathematical viewers. it truly is, notwithstanding, essentially meant for starting graduate and complicated undergraduate scholars encountering noncommutative algebra for the 1st time.
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Extra info for Introduction to Noncommutative Algebra (Universitext)
Example text
Its standard basis contains a unity 1 and seven elements whose squares equal −1. Although not associative, O satisfies weak associativity conditions x 2 y = x(xy) and yx 2 = (yx)x for all x, y ∈ O; nonassociative algebras satisfying these two conditions are called alternative. Further, O is a nonassociative division algebra, which, by definition, means that for all a, b ∈ O with a = 0, the equations ax = b and ya = b have unique solutions in O. Let this be enough in our attempt to make the reader curious and challenged for seeking further information about octonions elsewhere.
We say that an element x from an F-algebra A is algebraic if there exists a nonzero polynomial f (ω) ∈ F[ω] such that f (x) = 0. This is, of course, just the repetition of the definition given in Prerequisites for field extensions, and just as before we define the degree of algebraicity. If every element in A is algebraic, then A is said to be an algebraic algebra. 1. 5 Every finite dimensional algebra is an algebraic algebra. Proof The set of all powers of each element is linearly dependent. 6 1 Finite Dimensional Division Algebras Our proof of Frobenius’ theorem thus shows that an algebraic real division algebra is isomorphic to R, C, or H, and hence it is finite dimensional.
We will now present the most fundamental example. First we have to introduce some notation and terminology. Consider the matrix ring Mn (S) where S is an arbitrary unital ring. Let Eij denote the matrix whose (i, j) entry is 1 and all other entries are 0. We call Eij , 1 ≤ i, j ≤ n, the standard matrix units. By aEij we denote the matrix whose (i, j) entry is a ∈ S and other entries are 0. Every matrix in Mn (S) is a sum of such matrices. Note that for every A = (aij ) ∈ Mn (S) we have Eij AEkl = ajk Eil for all 1 ≤ i, j, k, l ≤ n.