By Rabib Islam
Read Online or Download Morita equivalence PDF
Similar algebra & trigonometry books
VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating rules 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 principles so successfully that the classical Wedderburn-Artin theorem is a straightforward end result, and furthermore, a similarity type [AJ within the Brauer team Br(k) of Azumaya algebras over a commutative ring okay involves all algebras B such that the corresponding different types mod-A and mod-B such as k-linear morphisms are similar by means of a k-linear functor.
The current monograph on matrix partial orders, the 1st in this subject, makes a different presentation of many partial orders on matrices that experience involved mathematicians for his or her good looks and utilized scientists for his or her wide-ranging program capability. aside from the LÃ¶wner order, the partial orders thought of are rather new and got here into being within the overdue Seventies.
Initially, my purpose used to be to put in writing a "History of Algebra", in or 3 volumes. In getting ready the 1st quantity I observed that during old civiliza tions geometry and algebra can't good be separated: increasingly more sec tions on old geometry have been additional. as a result the recent name of the publication: "Geometry and Algebra in historical Civilizations".
Additional resources for Morita equivalence
R1 sn ) + · · · + (0, . . , 0, 1) ⊗ (rn s1 , . . , rn sn ) = 0 =⇒ (r1 . . , rn ) ⊗ (s1 , . . , sn ) = 0. Since g˜ is Mn (R)-balanced, it is a bimodule homomorphism, and hence a bimodule isomorphism. 3. 16 from a bicategorical perspective. That is, we explore what it means for a bimodule to be faithfully balanced in Bim. 20. The map f : R → EndR (RR ) defined by f (r) = φr , where φr is left multiplication by r, is a ring isomorphism. Proof. We first show that all the endomorphisms of RR are left multiplications by elements of R.
Since c ∈ ker β2 , we know that c ∈ im α2 , so we can pick d ∈ T L such that α2 (d) = c. Since ψ L is surjective, we can pick e ∈ Q ⊗R L such that ψ L (e) = d. Since ψ F is injective, b is the unique preimage of c through ψ F . Since the diagram commutes, we know that α1 (e) = b. Finally, by exactness, a = β1 (b) = β1 (α1 (e)) = 0. Thus, ψ M is injective, and hence an isomorphism. We can now use our knowledge of Bim to prove the following corollary. 1]). The rings R and S are Morita equivalent if and only if there is an (R, S)-bimodule P and an (S, R)-bimodule Q such that R PS ⊗S S QR R RR and S QR ⊗R R PS S SS as bimodules.
Sn ) = (1, 0, . . , 0) ⊗ (r1 s1 , . . , r1 sn ) + · · · + (0, . . , 0, 1) ⊗ (rn s1 , . . , rn sn ). We thus also see that g˜ is injective: ˜ 1 , . . , rn ) ⊗ (s1 , . . , sn )) = 0 =⇒ ∀i, ∀j, ri sj = 0 g((r =⇒ (1, 0, . . , 0) ⊗ (r1 s1 , . . , r1 sn ) + · · · + (0, . . , 0, 1) ⊗ (rn s1 , . . , rn sn ) = 0 =⇒ (r1 . . , rn ) ⊗ (s1 , . . , sn ) = 0. Since g˜ is Mn (R)-balanced, it is a bimodule homomorphism, and hence a bimodule isomorphism. 3. 16 from a bicategorical perspective. That is, we explore what it means for a bimodule to be faithfully balanced in Bim.