By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes whilst there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's resolution organizes rules so successfully that the classical Wedderburn-Artin theorem is a straightforward end result, and furthermore, a similarity classification [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring okay contains all algebras B such that the corresponding different types mod-A and mod-B which include k-linear morphisms are similar through a k-linear functor. (For fields, Br(k) includes similarity periods of easy vital algebras, and for arbitrary commutative okay, this is often subsumed lower than the Azumaya [51]1 and Auslander-Goldman [60J Brauer crew. ) a variety of different situations of a marriage of ring idea and classification (albeit a shot­ gun wedding!) are inside the textual content. additionally, in. my try to additional simplify proofs, significantly to get rid of the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside of ring concept. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre­ spondence theorem for projective modules (Theorem four. 7) instructed by means of the Morita context. As a derivative, this gives starting place for a slightly whole concept of straightforward Noetherian rings-but extra approximately this within the creation.

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Conversely, if |Bi | = 1 for all i ∈ I \ {k}, then |Bk | = |A|, and hence the projection function πk is an isomorphism between A and B k by the pigeon-hole principle, because A is finite. So the conditions (i) and (ii) are equivalent. By essentially the same argument, if A ∼ = B × C, then B is trivial iff A ∼ = C, and C is ∼ trivial iff A = B. 57(ii). week 9 52 Neither of the conditions (ii) or (iii) of the corollary is equivalen to direct indecomposability for arbitrary Σ-algebras. In particular, it follows from the remarks at the end of the chapter that every countably infinite left-trivial semigroup satisfies both (ii) and (iii), but none of these algebras is directly irreducible.

For an important class of algebras only one iteration of the relative product is needed. 32. Let A be a Σ-algebra and let α, β ∈ Co(A). The following are equivalent. (i) α ; β ⊆ β ; α. (ii) α ; β = β ; α. (iii) α ; β ∈ Co(A). (iv) α ; β = α ∨Co(A) β. Proof. Here is the chain of implications we will prove: (i) ⇐⇒ (ii) =⇒ (iv) ⇐⇒ (iii) =⇒ (ii). (i) ⇐⇒ (ii). The implication from right to left is trivial. Assume α;β ⊆ β ;α. Since α and β are symmetric, α = α and β = β. Thus β ; α = β ; α = (α ; β) ⊆ (β ; α) = α ; β = α ; β.

So ∆A and ∇A α ˆ = ∇A ∩ α are the only factor congruences of A. (iii) =⇒ (i). Suppose A ∼ = i∈I B i . Let αi : i ∈ I be a factor congruence system ∼ such that B i = A/αi for each i ∈ I. By assumption each αi is either ∆A or ∇A . They all cannot be ∇A since otherwise each B i is trivial which is impossible since A is nontrivial. ˆi = j∈I\{i} αj . Note that, for So αk = ∆A for at least one k ∈ I. For each i ∈ I let α each i ∈ I \ {k}, α ˆ i ⊆ αk = ∆A and hence α ˆi = ∆A . But by Thm. 53 αi and α ˆ i are complementary congruences.

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