Associative Algebras by R.S. Pierce

By R.S. Pierce

For lots of humans there's existence after forty; for a few mathematicians there's algebra after Galois conception. the target ofthis ebook is to turn out the latter thesis. it really is written basically for college students who've assimilated colossal parts of a regular first yr graduate algebra textbook, and who've loved the event. the fabric that's awarded right here shouldn't be deadly whether it is swallowed by means of people who're no longer individuals of that staff. The items of our awareness during this booklet are associative algebras, commonly those which are finite dimensional over a box. This topic is perfect for a textbook that would lead graduate scholars right into a really expert box of analysis. the most important theorems on associative algebras inc1ude probably the most ultimate result of the good heros of algebra: Wedderbum, Artin, Noether, Hasse, Brauer, Albert, Jacobson, etc. the method of refine­ ment and c1arification has introduced the evidence of the gem stones during this topic to a degree that may be liked by means of scholars with simply modest historical past. the topic is sort of specific within the wide variety of contacts that it makes with different components of arithmetic. The learn of associative algebras con­ tributes to and attracts from such issues as staff concept, commutative ring conception, box concept, algebraic quantity concept, algebraic geometry, homo­ logical algebra, and class conception. It even has a few ties with components of utilized arithmetic.

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Let A(·) : Po (G) −→ Po (E(G)) be defined by A(·) = Hom(G, ·) ⊗End(G) End(G)/N (End(G)). 1. A(·) is an additive full functor that preserves direct sums. That is, given an E(G)-module map f : A(H) → A(H ) there is a group map φ : H → H such that A(φ) = f . 2. A(·) induces a bijection α : {(H) H ∈ Po (G)} −→ {(P ) P ∈ Po (E(G))} between the set of isomorphism classes (H) of H ∈ Po (G) and the set of isomorphism classes (P ) of P ∈ Po (A(G)). 3. A(·) induces a bijection between the set of quasi-isomorphism classes [H] of H ∈ Po (G) and the set of quasi-isomorphism classes [P ] of finitely generated projective right E(G)-modules.

We say that G has a unique decomposition if 1. G has an indecomposable decomposition G ∼ = G1 ⊕ · · · ⊕ Gt and 2. Given an indecomposable decomposition G ∼ = G1 ⊕ · · · ⊕ Gs then s = t and after a permutation of the subscripts, Gi ∼ = Gi for each i = 1, . . , t. In this case we call G1 ⊕ · · · ⊕ Gt the unique decomposition of G. The unique decomposition of an rtffr group G is necessarily indecomposable. Rtffr groups having unique decomposition are considered to be rare. 4 1. The Fundamental Theorem of Abelian Groups states that if p ∈ Z is a prime and if G is a finite p-group then G has a unique decomposition G = Z/pn1 Z ⊕ · · · ⊕ Z/pnt Z for some integers 0 < n1 ≤ · · · ≤ nt .

H ⊕K ∼ 3. If G(n) = = H ⊕ K for some integer n > 0 and some rtffr groups K and K then K ∼ =K. Proof: We begin the proof with some general comments about finitely generated projective right E(G)-modules. 4 E(G) = E(G1 ) × · · · × E(Gt ) where E(Gi ) = A(Gi ) is indecomposable for each i = 1, . . , t. 34 CHAPTER 2. MOTIVATION BY EXAMPLE Furthermore since E(Gi ) is a pid, given a finitely generated projective right E(G)-module P there are integers p1 , . . , pt ≥ 0 such that P ∼ = E(G1 )(p1 ) ⊕ · · · ⊕ E(Gt )(pt ) .

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