By Carl Faith

ISBN-10: 3540057056

ISBN-13: 9783540057055

Publication via religion, Carl

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**Extra info for Algebra II. Ring Theory: Ring Theory**

**Example text**

5. We say that two simple (not necessarily central) k-algebras A and B are Brauer equivalent if we have A ∼ =k Ms (D) for some =k Mr (D) and B ∼ integers r, s ≥ 1 and some division k-algebra D. We denote it by A ∼k B. In other words, two simple k-algebras are Brauer equivalent if they correspond to isomorphic division k-algebras via Wedderburn’s theorem. 8 (3). Moreover, if A and B are two Brauer equivalent central simple k-algebras, they have the same index. The following lemma will provide an easy way to produce examples of simple k-algebras which are Brauer equivalent to A.

Thus we have dimk (I) = dimL (I ⊗k L) = dimL (A ⊗k L) = dimk (A), and so I = A. Hence, A is simple. This concludes the proof. We will go back to the study of the centralizer at the end of this chapter. For the moment, we would like to end this section by introducing the opposite algebra of a central simple k-algebra A. 9. Let A be k-algebra, and let Aop be the set Aop = {aop | a ∈ A}. The operations k × Aop −→ Aop (λ, aop ) −→ (λ·a)op , Aop × Aop −→ Aop op op (aop 1 , a2 ) −→ (a1 + a2 ) 34 III.

4. 1. 2 since ϕH,C and √ √ ϕQ,L (L = k( a) or k( b)) are indeed suitable injective R-algebra and k-algebra morphisms respectively. 30 II. QUATERNION ALGEBRAS √ Notice that each coeﬃcient of a codeword in CQ,k(√b) is an element of k( b), which √ is a vector space of dimension 2 over k. Thus while an element of k( b) can be seen as one signal sent, if information symbols to be sent are chosen in k, we have that one signal actually contains two information symbols. Note here the important diﬀerence between a code based on general quaternion algebras and one based on Hamilton quaternions.

### Algebra II. Ring Theory: Ring Theory by Carl Faith

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