A CATEGORIES EQUIVALENCE OF ASSOCIATIVE BIMODULES

. In this paper we use the classical Wedderburn’s Kronecker Factorization Theorem to prove that category of bimodules over B and the category of bimodules over M n ( B ) are equivalent, where B is some unital associative algebra. In addition to this, we classify the irreducible bimodules over M n ( F )


Introduction
An associative algebra is an F -vector space A with a bilinear binary operation (x, y) → xy satisfying the following identity: (x, y, z) = 0, where (x, y, z) = (xy)z − x(yz) is the associator of the elements x, y, z ∈ A.
The description of the structure of algebras and superalgebras that contain certain finite-dimensional algebras and superalgebras has a rich history, which has important applications in representation theory and category theory (for example, see [2,3,6,7,8,9,10,11,12,14]).The classical Wedderburn Theorem says that if a unital associative algebra A contains a central simple subalgebra of finite dimension B with the same identity element, then A is isomorphic to a Kronecker product S ⊗ F B, where S is the subalgebra of the elements that commute with each b ∈ B.
In particular, if A contains M n (F ) as a subalgebra with the same identity element, we have A ∼ = M n (S) "coordinated" by S. Kaplansky in Theorem 2 of [5]  VICTOR L ÓPEZ SOL ÍS AND MARLENNHI MORENO VILLANUEVA B. Jacobson in Theorem 1 of [2] gave a new proof of the result of Kaplansky using his classification of completely reducible alternative bimodules over a field of characteristic different of 2 and finally V. López-Solís in [8] proved that this result is valid for any characteristic.Using this result, Jacobson [2] proved a Kronecker Factorization Theorem for Jordan algebras that contain the Albert algebra with the same identity element.The statements of this type are usually called Kronecker factorization theorems.
In [13], K. McCrimmon says that Wedderburn's Kronecker Factorization Theorem (KFT) is the grandfather of all Kronecker factorization theorems.Despite its great importance, we certainly have not found bibliographical references of this result in ring representation theory or algebras.Motivated by this lack, we thought it would be useful to describe some applications of the KFT and thus see its utility.
In this note we use the KFT to prove that the category of bimodules over B and the category of bimodules over M n (B) are equivalent, where B is some unital associative algebra.In addition to this, we classify the irreducible bimodules over M n (F ).

Preliminaries
Let A be an associative algebra over F .A vector space V over F is called an A-bimodule if there are bilinear mappings A × V → V and V × A → V sending (a, v) to av and (v, a) to va, respectively.We say that V is an associative bimodule for A if the algebra E = A ⊕ V with the multiplication given by for all a, b ∈ A and v ∈ V , where (x, y, z) := (xy)z − x(yz) is the associator of x, y and z.Therefore this definition of associative bimodule coincides with the usual one.
Suppose A has an identity element 1, then the associative bimodule V is called a unital associative bimodule for A if 1v = v1 = v for all v ∈ V.For the definition of unital right modules, see [4].
Let us recall some elementary facts about matrix units in M n (R), where R is a ring with identity.For i, j = 1, . . ., n, we define e ij as a matrix whose entry (i, j) Proposition 2.1 is proved in the Jacobson's book (see Proposition 1.4 in [4]).Our aim is to prove this result for bimodules using the famous KFT.
Above we have defined the e ij as the matrices with 1 in the input (i, j) and 0 in the others.We call the set of these elements a system of unitary matrices that satisfy where δ is the Kronecker delta.
As mentioned in the Introduction, the KFT points out a very interesting feature of central simple algebras, namely, whenever they sit in a larger algebra they do so in a very particular way.In fact the property of the theorem characterizes finite we have In particular: Corollary 2.3.Let A be an algebra with identity element 1 such that A contains a system of n 2 unitary matrix elements.Then A ∼ = M n (B), where B is the subalgebra of the elements that commute with each e ij of the system.
Denote by F a field of arbitrary characteristic.Corollary 2.3 says that if A contains M n (F ) as a subalgebra with the same identity element, then A ∼ = M n (B), that is, A is "coordinated" by B and acquires the matrix structure of M n (F ).

A categories equivalence
Next, we state and prove the most important results of the article.The first result is an equivalence of categories.Indeed, it is an analogue of Proposition 2.1 for bimodules and it is given using the KFT.The second application of KFT is related to the classification of irreducible bimodules.
Let B be an arbitrary unital associative algebra over the base field F .Denote by Proof.We want to prove that generalized the Wedderburn result to the alternative algebras A and the split Cayley algebra The paper is a part of the "Tesis de Licenciatura" at the National University of Santiago Antúnez de Mayolo of the second author.The first author was supported by Resolution of the Organizing Commission N • 629-2022-UNAB of the National University of Barranca.Also the authors gratefully acknowledge financial support by CONCYTEC-PROCIENCIA within the framework of the call "Proyecto Investigación Básica 2019-01" [380-2019-FONDECYT].

3 is 1 Proposition 2 . 1 .
and the other entries are 0. Also, we have the multiplication table e ij e kl = δ jk e il and e ii = 1.Hereinafter right modules will mean unital right modules.Similarly, bimodules will mean unital associative bimodules.Denote by mod − R the category of right modules for a fixed ring R. Ob(mod − R) is the class of right modules for R and the morphisms are R-module homomorphisms.Products are composites of maps.Similarly, mod − M n (R) denote the category of right modules over M n (R).Let R be a ring and M n (R) be the ring of matrices of order n×n with entries in R. Then the categories mod − R and mod − M n (R) of modules to the right over R and M n (R) respectively, are equivalent.
dimensional central simple algebras.See the Herstein's book (see Theorem 4.4.2 in [1]).Theorem 2.2.Let A be a unital associative algebra that contains a central simple subalgebra of finite dimension S with the same identity element, then A is isomorphic to a Kronecker product S ⊗ F B, where B is the subalgebra of the elements that commute with each b ∈ S. Define [a, b] := ab − ba the commutator of the elements a, b ∈ A. In Theorem 2.2

Theorem 3 . 1 .
Bimod−B the category of bimodules for B, where Ob(Bimod−B) is the class of bimodules over B. • Bimod−M n (B) the category of bimodules for M n (B), where Ob(Bimod− M n (B)) is the class of bimodules over M n (B).The categories Bimod−B and Bimod−M n (B) are equivalent.

Let N ∈ fCorollary 3 . 2 .
Ob(Bimod−B) and consider the split null extension E = B ⊕ N of the associative algebra B by the bimodule N .Since E is an associative algebra we can form the matrix algebra K = M n (E) containing M n (B) as a subalgebra.Thus, K contains the ideal M = M n (N ) ∩ K = M n (N ) which is the set of matrices of K whose entries are in the ideal N of E. Consequently, M is a bimodule for M n (B) relative to the multiplication defined in M n (E).Therefore, M ∈ Ob(Bimod−M n (B)) and will be the M n (B)-bimodule associated with the given bimodule N of B. Thus, we have a mapN → M = M n (N ) of Ob(Bimod−B) to Ob(Bimod−M n (B)).If f : N −→ N ′ is a B-bimodule homomorphism, (a n1 ) . . .f (a nn ) of M n (B)-bimodules of M to M ′ .The maps N → M = M n (N ) and f → f constitute a functor T : Bimod−B −→ Bimod−M n (B).If N ∈ Ob(Bimod−B), denote M = T(N ).Since E = B ⊕ N , we have that K = M n (B) ⊕ M .The fact that N 2 = 0 in E implies that M 2 = 0 in K, so K is the split null extension of M n (B) by its bimodule M .It can be easily verified that T is actually a functor of the category Bimod−B to the category Bimod−M n (B).Furthermore, for every pair of objects N and N ′ of Bimod−B, the following equality holdsT(hom(N, N ′ )) = hom(T(N ), T(N ′ )).Thus, N and N ′ are isomorphic if and only if T(N ) and T(N ′ ) are isomorphic.Similarly, the functor T offers a lattice isomorphism of the lattice of the submodules of N relative to B on the lattice of the submodules of M over M n (B).To complete the reduction of the theory of bimodules for M n (B) to that of bimodules for B, we will show that each M n (B)-bimodule is isomorphic to some bimodule associated with a bimodule for B. Consider a bimodule V for M n (B) and letA = M n (B) ⊕ Vbe the split null extension of M n (B) by V .Thus A is an associative algebra (with identity element 1, the identity of M n (B)) containing the matrix algebra M n (F ) ⊆ M n (B) as a unital subalgebra, then by Corollary 2.3 of KFT, there exists a unital associative algebra D such that A = M n (D), thenM n (D) = M n (B) ⊕ V.Let W be the set of elements of D that appear in the entries of the matrices of V .ThenV := M n (W ), where W ◁ D and W 2 = 0 in D, since V ◁ A and V 2 = 0 in A; so D = B ⊕ Wis the split null extension of B by its bimodule W , then W is a bimodule over B.Thus T(W ) = V.□ Every bimodule V over B is completely reducible if and only if T (V ) is completely reducible over M n (B).