By Rabib Islam

**Read Online or Download Morita equivalence PDF**

**Similar algebra & trigonometry books**

**Algebra. Rings, modules and categories**

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.

**Matrix Partial Orders, Shorted Operators and Applications (Series in Algebra)**

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.

**Geometry and Algebra in Ancient Civilizations**

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**

**Example text**

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.