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Quasi-Frobenius and separable ring extensions.

Abstract

The purpose of this paper is to study some quasi-Frobenius and separable ring extensions. In particular, we extend slightly some results of Kitamura, further, we show that H-separable (separable) extension of rings hold for some ring endomorphisms.

Keywords: Quasi-Frobenius extension, H-separable (separable) extension, bimodules, projective.

1. Introduction

The purpose of this work is to study some quasi-Frobenius and separable ring extensions. In [4] (resp: [2]) Y. Miyashita (resp: Y. Kitamura) developed a commutator theory of Frobenius (resp: quasi-Frobenius) ring extensions.

In this note, we extend slightly Kitamura's [2, Theorem 1.1 and Proposition 1.2]. Further, we show that H-separable (separable) extension of rings hold for some endomorphism of rings. The main result of the present paper is the following theorem:

Theorem 1.1. Let [rho] : B [right arrow] A be a ring homomorphism and [U.sub.A] a right A-module

1. Assume that [[bar.U].sub.A] = [Hom.sub.B]([sub.A][A.sub.B], [U.sub.B]) \ [U.sub.A]. If [rho] : B [right arrow] A is H-separable (resp: separable) then so is j : [LAMBDA] [right arrow] [GAMMA] defined via: j([lambda])(f) = [lambda] [??] f, for [lambda] [member of] [LAMBDA] and f [member of] [bar.U] where [LAMBDA] = End([U.sub.B]) and [GAMMA] = End([[bar.U].sub.A])

2. Assume that [[??].sub.A] = [U.sub.B] [cross product] [A.sub.A]\[U.sub.A]. If [rho] : B [right arrow] A is H-separable (resp: separable) then so is [j.sup.*] : [LAMBDA] [right arrow] [[LAMBDA].sup.*] defined via: [j.sup.*]([lambda]) = [lambda][cross product][Id.sub.A], [lambda] [member of] [LAMBDA], where [LAMBDA] = End([U.sub.B]) and [[GAMMA].sup.*] = End([[??].sub.A])

Throughout this paper, for module theoretic notions and notations we will follow [1, 2] and [3]. In the following, we recall some definitions and notations that will be retained in this note.

All rings are associative with identity 1 (if A is a ring, we assume that 1 [not equal to] 0), all modules are unital and all ring extensions contain the common identity. The homomorphisms of modules operate at the opposite side of the scalar and the ring homomorphisms send identity element to identity element.

In this paper, we shall denote by End([M.sub.A]), the A-endomorphism ring of a right A-module M and consider End([M.sub.A]) as a left operator domain of M. The endomorphism ring of M considered as a End([M.sub.A])-module is called the biendomorphism ring of [M.sub.A] and it is abbreviated by: BiEnd([M.sub.A]) = End([sub.T]M) where T = End([M.sub.A]).

Let [sub.R][M.sub.S] and [sub.R][N.sub.S] be two (R, S)-bimodules. The notation [sub.R][M.sub.S]\[sub.R][N.sub.S] means that M is isomorphic to a direct summand of a direct sum of finitely many copies of N as a (R, S)-bimodules. We denote by [sub.R][M.sub.S] ~[sub.R] [M.sub.S] the fact that [sub.R][M.sub.S]\[sub.R][N.sub.S] and [sub.R][N.sub.S]\[sub.R][M.sub.S]. The previous notation holds if M and N are right S-module or left R-module. It is well known that a module [M.sub.S] is finitely generated (f.g: briefly) and projective [resp: a generator, a progenerator] if and only if [M.sub.S]\[S.sub.S] [resp: [S.sub.S]\[M.sub.S], [M.sub.S] ~ [S.sub.S]].

A ring homomorphism [rho] : B [right arrow] A is called a ring extension. If [rho] : B [right arrow] A is injective, the ring extension is denoted by A/B. If [rho] : B [right arrow] A is a ring extension, then every A-module may be regarded as a B-module via [rho] and every A-linear morphism is also B-linear. Let [rho] : B [right arrow] A and [rho]' : B' [right arrow] A' be ring homomorphisms, [.sub.A][U.sub.A]' and [sub.A][V.sub.A'] be two bimodules such that [sub.A][U.sub.A'] [??] [sub.A] [V.sub.A'], then U and V are also (B,B')-bimodules such that [sub.B][U.sub.B'] [??] [sub.B] [V.sub.B'] canonically.

A ring homomorphism [rho] : B [right arrow] A is called Frobenius extension (denoted by: FE) if [A.sub.B]\[B.sub.B] and [sub.B][A.sub.A] [??] [Hom.sub.B]([sub.A][A.sub.B],[sub.B] [B.sub.B]). If [A.sub.B]\[B.sub.B] and [sub.B][A.sub.A] [??] [Hom.sub.B]([sub.A][A.sub.B],[sub.B] [B.sub.B]), then the ring extension [rho] : B [right arrow] A is called right quasi-Frobenius extension (denoted by: QFE). Symmetrically, if [sub.B]A\[sub.B]B and [sub.A][A.sub.B]\[Hom.sub.B]([sub.B][A.sub.A],[sub.B] [B.sub.B]), then [rho] : B [right arrow] A is called left quasi-Frobenius extension. A right and left QFE is called a QFE one (cf. Kitamura [3]). A ring extension [rho] : B [right arrow] A is called H-separable (resp: separable) if: [sub.A][A.sub.B] [cross product] [A.sub.A]\[sub.A][A.sub.A] [resp: [sub.A][A.sub.A]\[sub.A][A.sub.B] [cross product] [A.sub.A]].

Remark 1.2. It is easily seen that, [rho] : B [right arrow] A is a left QFE if and only if (iff: briefly) [A.sub.B]\[B.sub.B], [sub.B]A\[B.sub.B] and [Hom.sub.B]([sub.A][A.sub.B],[sub.B] [B.sub.B])\[sub.B][A.sub.A]. Hence, if [rho] : B [right arrow] A is a left or right QFE then A is finitely generated and projective as left and right B-module.

2. Preliminaries

We shall recall the following Miyashita's lemma which will be used in the proof of some results in this note.

Lemma 2.1. (Miyashita's Lemma 3.6 of [4]) Let A be a ring and B it's subring. Let [X.sub.A] and [Y.sub.A] be A-module such that [X.sub.A]\[Y.sub.A]. Then, there exist finitely many module homomorphisms [f.sub.i] : X [right arrow] Y and [g.sub.i] : Y [right arrow] X, 1 [less than or equal to] i [less than or equal to] n, such that [n.summation over (1)] [g.sub.i] [??] [f.sub.i] = i[d.sub.X].

Furthermore, let Z be a right B-module, C = End([X.sub.A]), D = End([Y.sub.A]) and F = End([Z.sub.B]), then we have the following:

(1) [Hom.sub.A](Y,X) [cross product][sub.D] [Hom.sub.B](Z, Y) [??] [Hom.sub.B](Z,X) as a (C, F)-bimodule via: g [cross product] k [right arrow] g [??] k. The inverse of this isomorphism is the mapping: h [right arrow] [n.summation over (1)] [g.sub.i] [??] ([f.sub.i] [??] h)

(2) [Hom.sub.B](Z,X) [??] [Hom.sub.D][Hom.sub.A](X, Y),[sub.D] [Hom.sub.B](Z, Y)] as a (C, F)-bimodule via:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The inverse of this isomorphism is the mapping: w [right arrow] [n.summation over (1)] [g.sub.i] [??] [w([f.sub.i])]

Lemma 2.2. (cf. [3]) Let [rho] : B [right arrow] A be a ring homomorphism and [U.sub.A] be a right A-module. Put [summation] = End[U.sub.A], [LAMBDA] = End[U.sub.B], [GAMMA] = End[[bar.U].sub.A] and [[GAMMA].sup.*] = End[[??].sub.A], where [[bar.U].sub.A] = [Hom.sub.B]([sub.A][A.sub.B], [U.sub.B]) and [[??].sub.A] = [U.sub.B] [cross product] [A.sub.A].

1. We have the following ring homomorphisms:

(i) i : [summation] [right arrow] [LAMBDA], the canonical injection

(ii) j : [GAMMA] [right arrow] [GAMMA] defined via: j([lambda])(f) = [lambda] [??] f, [lambda] [member or] [LAMBDA], f [member of] [bar.U], where [LAMBDA] = End([U.sub.B]) and [GAMMA] = End([[bar.U].sub.A])

(iii) [j.sup.*] : [LAMBDA] [right arrow] [[GAMMA].sup.*] defined via: [j.sup.*]([lambda]) = [lambda] [cross product] [Id.sub.A], [lambda] [member of] [LAMBDA], where [LAMBDA] = End([U.sub.B]) and [[GAMMA].sup.*] = End([[??].sub.A])

2. (a) If [A.sub.B] is a generator then j is injective.

(b) If [sub.B]A is a generator then [j.sup.*] is injective.

Proof.

1. It is note difficult to see that i, j and [j.sup.*] are ring homomorphisms. The fact that i is injective is obvious

2. (a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since [A.sub.B] generates [U.sub.B] then, for every u [member of] U, there exist finite many elements, [f.sub.1],..., [f.sub.n] in [bar.U] and [a.sub.1],..., [a.sub.n] in A such that u = [[summation].sup.n.sub.i=1][f.sub.i]([a.sub.i]). Thus, [lambda](u) = [[summation].sup.n.sub.i=1]([lambda] [??] [f.sub.i])([a.sub.i]). Using the fact that [lambda][[f.sub.i]([a.sub.i])] = 0, [for all]i, 1 [less than or equal to] i [less than or equal to] n, we have [lambda](u) = 0, [for all]u [member of] U. Hence [lambda] = 0, which means that j is injective.

(b) Recall that the tensor annihilator (cf. [1]) in a B-module [U.sub.B] of a B-module [sub.B]K is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[sub.B]K is said to be completely faithful in case [Ann.sub.U](K) = 0 for every right B-module U. It is easy to see that a generator is completely faithful. Since [sub.B]A is a generator by assumption, then [lambda] [cross product] [Id.sub.A] = 0 implies [lambda] = 0. Hence [j.sup.*] is injective.

Lemma 2.3. Let [rho] : B [right arrow] A be a ring homomorphism and [U.sub.A] a right A-module. Put [summation] = End[U.sub.A], [LAMBDA] = End[U.sub.B], [[bar.U].sub.A] = [Hom.sub.B]([sub.A][A.sub.B], [U.sub.B]) and [[??].sub.A] = [U.sub.B] [cross product] [A.sub.A].

1. Assume that: [[??].sub.A]\[U.sub.A]

(a) Then [Hom.sub.[summation]]([sub.[summation]][[LAMBDA].sub.[LAMBDA]], [.sub.[summation] [U.sub.A]) [??][sub.[LAMBDA] [[??].sub.A].

(b) Moreover, if [sub.B]A\[sub.B]B [resp: [sub.B]B\[sub.B]A] then [Hom.sub.[summation]]([sub.[summation]][[LAMBDA].sub.[LAMBDA]], [sub.[summation]]U))\[sub.[LAMBDA]]U

[resp: [sub.[LAMBDA]]U\[Hom.sub.[summation]]([sub.[summation]] [[LAMBDA].sub.[LAMBDA]],[sub.[summation]]U)].

2. Assume that: [[bar.U].sub.A]\[U.sub.A]

(a') Then [sub.[LAMBDA]][LAMBDA] [cross product][sub.[summation]] [U.sub.A] [??][sub.[LAMBDA]] [[bar.U].sub.A]

(b') Moreover, if [A.sub.B]\[B.sub.B] [resp: [B.sub.B]\[A.sub.B]] then [sub.[LAMBDA]],[[LAMBDA]sub.[summation]][cross product][sub.[summation]]U)\ [sub.[LAMBDA]]U [resp: [sub.[LAMBDA]]U\[sub.[summation]] [[LAMBDA].sub.[LAMBDA]] [cross product] [sub.[summation]]U].

Proof.

1. (a) We have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([alpha])

Since [[??].sub.A]\[U.sub.A], applying Miyashita's lemma, we have the following isomorphism:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

defined via:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus [phi] induces an additive and bijective morphism:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

It is not difficult to see that [phi] is ([LAMBDA],A) linear, then we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([beta]) as desired.

(b) If [sub.B]A\[sub.B]B [resp: [sub.B]B\[sub.B]A] then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using ([beta]) we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

as desired.

2. (a') We have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([gamma])

Since [[bar.U].sub.A]\[U.sub.A], applying Miyashita's lemma, we have the following isomorphism of ([GAMMA],A) bimodules where [GAMMA] = End([[bar.U].sub.A]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

defined via:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus we have an additive and bijective morphism:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is easy to see that [[bar.[psi]] is ([LAMBDA],A) linear. Hence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Similarly to (1.a), it follows from ([gamma]) and ([gamma]') that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

as desired.

(b') If [A.sub.B]\[B.sub.B] [resp:[B.sub.B]\[AvB]] then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using ([lambda]) we have: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] as desired. Now the proof of Lemma 2.3 is complete.

Lemma 2.4. Let [U.sub.A] be a right A-module. If [rho] : B [right arrow] A is a left QFE (resp: QFE, FE) then [[bar.U].sub.A] = [Hom.sub.B] ([sub.A][A.sub.B], [U.sub.B])\(resp :~,[??])[U.sub.B] [cross product] [A.sub.A] [??] [[??].sub.A]

Proof. If [rho] : B [right arrow] A is a left QFE (resp: QFE, FE) then [A.sub.B]\[B.sub.B], and [Hom.sub.B]([sub.A][A.sub.B],[sub.B] [B.sub.B])\ (resp:~,[??])[sub.B][A.sub.A].

Hence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since [A.sub.B] is f.g and projective, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

. Thus: [Hom.sub.B]([sub.A][A.sub.B], [U.sub.B])\(resp :~,[??])[U.sub.B] [cross product] [A.sub.A]

3. Ring Extensions

In the following proposition, we extend [2, Proposition 1.2]. In particular, first, we give a new version without the condition "[A/B, [phi]] is a QF extension" which is superfluous. On the other hand, the results of [2, Theorem 1.1] will not be used in the proof of this proposition.

Proposition 3.1. Let [rho] : B [right arrow] A be a ring homomorphism and [U.sub.A] be a right A-module. Put [summation] = End([U.sub.A]) and [LAMBDA] = End[U.sub.B]), then we may assume that [summation] is a subring of [LAMBDA].

(1) If [rho] : B [right arrow] A is H-separable then [sub.[summation]][[LAMBDA].sub.[summation]]\[sub.[summation] [[summation].sub.[summation]]

(2) If [rho] : B [right arrow] A is separable then [sub.[summation]][[summation].sub.[summation]]\[sub.[summation]] [[LAMBDA].sub.[summation]]

Assume that [[bar.U].sub.A] = [Hom.sub.B]([sub.A][A.sub.B],[U.sub.B])\ [U.sub.A] or [[??].sub.A] = [U.sub.B] [cross product] [A.sub.A]\[U.sub.A].

(3) If [sub.B][A.sub.B]\[sub.B][B.sub.B] then [LAMDA]/[summation] is H-separable

(4) If [sub.B][B.sub.B]\[sub.B][A.sub.B] then [LAMBDA]/[summation] is separable.

Proof.

(1) and 2): We have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (a)

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (b)

It follows from (a) and (b) that: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3)and 4): first, we have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (a')

On the other hand, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (b')

Now, we are going to prove the results desired with each of the two assumptions of this proposition.

First case: Assume that [[bar.U].sub.A]\[U.sub.A]. By (a') and (b'), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (c')

Then, applying Miyashita's Lemma, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (c")

as ([GAMMA], [[GAMMA].sup.*]) bimodules where: [GAMMA] = End [[bar.U].sub.A], [[GAMMA].sup.*] = End[[??].sub.A]. Hence, it follows from (c'), (c") and (1) of Lemma 2.2 that:

[[LAMBDA].sub.[summation]] [cross product][sub.[summation]] [LAMBDA] [??] [Hom.sub.A]([[??].sub.A], [[bar.U].sub.A]) (d')

as ([LAMBDA], [LAMBDA]) bimodules. We have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (e')

If [sub.B][A.sub.B]\[sub.B][B.sub.B] [resp: [sub.B][B.sub.B][sub.B][A.sub.B]] then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (f')

It follows from (e') and (f') that: [sub.[LAMBDA]] [[LAMBDA].sub.[summation]] [cross product][summation] [[LAMBDA].sub.[LAMBDA]]\[sub.[LAMBDA]] [[LAMBDA].sub.[LAMBDA]] [resp : [sub.[LAMBDA]][[LAMBDA].sub.[LAMBDA]]\[sub.[LAMBDA]] [[LAMBDA].sub.[summation]] [cross product][sub.[summation]] [[LAMBDA].sub.[LAMBDA]] which means that [GAMMA]/[summation] is H-separable [resp: separable] as desired.

Second case: Assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. By (b'), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (g')

[U.sub.A]\[U.sub.A] implies that [[summation].sub.P][sub.[summation]][summation] which means that [sub.[summation]]P is f.g and projective. Hence, we have the following isomorphism:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (g")

So, it follows from (g') and (g") that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (g'")

Further, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (h')

Since [[??].sub.A]\[U.sub.A], applying Miyashita's lemma, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (h")

as a ([[GAMMA].sup.*], A) bimodules where: [[GAMMA].sup.*] = End[[??].sub.A] and A [??] End([A.sub.A]). Since there exist ring homomorphisms [rho] : B [right arrow] A and [j.sup.*] : [LAMBDA] [right arrow] [[GAMMA].sup.*] then (h') and (h") yield that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

as a ([LAMBDA], B) bimodules. Thus:

[Hom.sub.[summation]]([sub.[summation]][P.sub.[LAMBDA]],[sub.[summation]] [U.sub.B]) [??] [sub.[LAMBDA]] [[??].sub.B] (k')

It follows from (g'") and (k') that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (l')

If [sub.B][A.sub.B]\[sub.B][B.sub.B] [resp: [sub.B][B.sub.B][sub.B][A.sub.B]] then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (m')

Now, (l') and (m') yields that: [sub.[LAMBDA]][[LAMBDA.sub.[summation]] cross product][sub.[summation]] [[LAMBDA].sub.[LAMBDA]]\[sub.[LAMBDA]] [[LAMBDA].sub.[LAMBDA]] [resp : [sub.[LAMBDA]][[LAMBDA].sub.[LAMBDA]][sub.[LAMBDA]][[LAMBDA].sub.[summation]] [cross product][sub.[summation]] [[LAMBDA].sub.[LAMBDA]]] which means that [LAMBDA]/[summation] is H-separable [resp: separable] as desired. Thus our proof is complete.

Lemma 3.2. Let [rho] : B [right arrow] A be a ring homomorphism and [U.sub.A] a right A-module. Put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Assume that: [sub.[LAMBDA]][U.sup.*] = [Hom.sub.[summation]]([sub.[summation]][[LAMBDA].sub.[LAMBDA]], [[sub.[summation]]U))\[sub.[LAMBDA]U

Then [sub.[LAMBDA]]U [cross product] [sub.[OMEGA]] [[omega].sub.[omega]] [??][sub.[LAMBDA]] [U.sup.*.sub.[omega]] = [Hom.sub.[summation]] ([sub.[summation]][[LAMBDA].sub.[LAMBDA]],[[sub.[summation]] [U.sub.[omega]]).

Proof. (This proof is similar to those of Lemma 2.3)

Since [sub.[LAMBDA]][U.sup.*]\[sub.[LAMBDA]]U, applying Miyashita's lemma, we have the following ([LAMBDA], C) isomorphism of bimodules where C = End([sub.[LAMBDA][U.sup.*]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

defined via:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus we have an additive and bijective morphism:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is easy to see that [bar.v] is ([LAMBDA], [omega]) linear. Hence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, we have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] as desired:

The next theorem extends slightly [2, Theorem 1.1] (See also [3] and [5] for the proof). In fact, we change the condition "A/B is a QFE" by the condition of Lemma 2.4, which is less restrictive.

Theorem 3.3. Let [rho] : B [right arrow] A be a ring homomorphism and [U.sub.A] a right A-module. Assume that the following two conditions hold:

(1) [[??].sub.A] = [U.sub.B] [cross product] [A.sub.A]\[U.sub.A]

(2) [[bar.U].sub.A] = [Hom.sub.B]([sub.A][A.sub.B],UB)\(resp :_~,_[??]) [U.sub.B] [cross product] [A.sub.A] = [[??].sub.A]

where [LAMBDA] = End([U.sub.B])

Put [summation] = End([U.sub.A]). Then we have the following:

(1*) [LAMBDA]]/[summation] is a left QFE (resp: QFE, FE)

(2*) Moreover, if [sub.B]A is f.g and projective, we have:

(a) [Hom.sub.[summation]]([sub.[summation]][[LAMBDA].sub.[LAMBDA]], [[sub.[summation]]U)

(b) [omega]/[OMEGA] is a left QFE (resp: QFE, FE) where [omega] = BiEnd([U.sub.A]) = End([sub.[summation]]U) and [OMEGA] = BiEnd([U.sub.B]) = End([sub.[LAMBDA]]U)

Proof.

(1*) We have the following isomorphisms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1*[alpha])

Since [[??].sub.A]\[U.sub.A], then it follows from (1*[alpha]) that [sub.[summation]][LAMBDA]\[Hom.sub.A]([sub.B][A.sub.A],[sub.[summation]] [U.sub.A]) =[sub.[summation]] [summation] which means that [sub.[summation]][LAMBDA] is f.g and projective (1*[alpha]')

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1*[beta])

Thus, using the second assumption, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1*[beta]')

Since [[??].sub.A]\[U.sub.A], it follows from Lemma 2.3 that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1*[gamma])

Using (1*[beta]') and (1*[gamma]) we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1*[lambda])

It follows from (1*[alpha]') and (1*[lambda]) that [LAMBDA]/[summation] is a left QFE (resp: QFE, FE)

(2*) (a) follows from (1b) of Lemma 2.3.

(b) We have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b[[alpha])

So, it follows from (2*a) and (2*b[alpha]) that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which means that [sub.[OMEGA]][omega] is f.g and projective (2*b[alpha]')

First we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b[beta])

On the other hand, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b[beta]')

Since [LAMBDA]/[summation] is a left QFE by (1*) then [[LAMBDA].sub.[summation]] is also f.g and projective (see the remark in the introduction). Thus (2*b[beta]') implies that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b[gamma])

By (1*), [LAMBDA]/[summation] is a left QFE (resp: QFE, FE) then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b[gamma]')

So, it follows from (2*b[gamma]) and (2*b[gamma]') that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b[gamma]")

Hence,(2*b[beta]) and (2*b[gamma]") yield that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b,)

Since [Hom.sub.[summation]]([sub.[summation]][[LAMBDA].sub.[LAMBDA]], [[sub.[summation]]U) by (2*a) of this theorem, then by Lemma 3.2, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b[delta])

It follows from (2*b[lambda]) and (2*b[delta]) that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2*b[mu])

(2*b[alpha]') and (2*b[mu]) show that: [omega]/[OMEGA] is a left QFE (resp: QFE, FE) as desired. End of the proof.

The following theorem is the main result of this paper.

Theorem 3.4. Let [rho] : B [right arrow] A be a ring homomorphism and [U.sub.A] be a right A-module.

1. Assume that [[bar.U].sub.A] = [Hom.sub.B]([sub.A][A.sub.B],UB) [U.sub.A]. If [rho] : B [right arrow] A is H-separable (resp: separable). Then so is j : [LAMBDA] [right arrow] [GAMMA] defined via: j([lambda])(f) = [lambda] [??] f, [lambda] member of] [LAMBDA], f [member of] [bar.U] where [LAMBDA] = End([U.sub.B]) and [GAMMA] = End([[bar.U].sub.A])

2. Assume that [[??].sub.A] = [U.sub.B] [cross product] [A.sub.A][U.sub.A]. If [rho] : B [right arrow] A is H-separable (resp: separable). Then so is [j.sup.*] : [LAMBDA] [right arrow] [[LAMBDA].sup.*] defined via: [j.sup.*]([lambda]) = [lambda] [Id.sub.A], [lambda] [member of] [LAMBDA], where [LAMBDA] = End([U.sub.B]) and [[GAMMA].sup.*] = End([[??].sub.A])

Proof.

1. We have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1a)

Since [[bar.U].sub.A]\[U.sub.A], then [[bar.U].sub.B]\[U.sub.B]. Hence, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus [sub.[LAMBDA]][GAMMA] is f.g and projective (1a'). Furthermore, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1b)

Since [sub.[LAMBDA]][GAMMA] is f.g and projective, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1b')

Now, (1b) and (1b') yield that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1b")

Using (1a) we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1c)

Since [[bar.U].sub.B]\[U.sub.B], Miyashita's lemma implies that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1c')

as (T, S) bimodules where T = End([[bar.U].sub.B]) and S = End ([A.sub.B]).

We may assume that [GAMMA] = End([[bar.U].sub.A]) is a subring of T = End([[bar.U].sub.B]). Let [??] : A [right arrow] S defined via: a [right arrow] [[??].sub.a] : x [right arrow] ax be the left multiplication. Then [??] is a ring monomorphism, thus A is embedded in S as a subring by [??]. Hence the last (T, S) isomorphism (cf: (1c')) is also a ([LAMBDA], A) isomorphism. Then we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1d)

If [rho] : B [right arrow] A is H-separable (resp: separable) then:

[sub.A][A.sub.B] [cross product] [A.sub.A]\[sub.A][A.sub.A][resp :[sub.A] [A.sub.A]\[sub.A][A.sub.B] [cross product] [A.sub.A]]

which implies that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1e)

It follows from (1d) and (1e) that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which means that j : [LAMBDA] [right arrow] [GAMMA] is H-separable [resp: separable] as desired.

2. We have the following isomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2a)

Since [[??].sub.A]\[U.sub.A] then [[??].sub.B]\[U.sub.B]. Hence, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus [[GAMMA].sup.*.sub.[LAMBDA]] is f.g and projective (2a'). Furthermore, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2b)

Since [sub.[LAMBDA]][GAMMA] is f.g and projective, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2b')

Now, (2b) and (2b') yield that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2b")

Using (2a) we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2c)

Since [[??].sub.B]\[U.sub.B], Miyashita's lemma implies that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2c')

as (D, B) bimodules where D = End([[??].sub.B]) and B [??] End([B.sub.B]). We may assume that [[GAMMA].sup.*] = End([[??].sub.A]) is a subring of D = End([[??].sub.B]). Then, the last (D, B) isomorphism (cf: (2c')) is also a ([[GAMMA].sup.*], B) isomorphism. Hence we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2c")

So it follows from (2b") and (2c") that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2d)

Clearly, as above, it follows from (2d) that if [rho] : B [right arrow] A is H-separable (resp: separable) then so is [j.sup.*] : [LAMBDA] [right arrow] [[GAMMA].sup.*]. Thus the proof of the main theorem is completely achieved.

References

[1] Anderson F.W. and Fuller K.R., 1975, Rings and categorie of modules. Springer-Verlag New York.

[2] Kitamura Y., 1975, Centralizers of a module over a quasi-Frobenius extension, Math J. Okayama. Univ., 17, pp. 103-123.

[3] Kitamura Y., 1981, Quasi-Frobenius extension with Morita duality, Journal of Algebra, 73, pp. 275-286.

[4] Miyashita Y., 1970, On Galois extension and crossed products, J. fac. Sci. Hokkaido Univ. Ser. I, 21, pp. 97-121.

[5] Sow D. and Sanghare M., 2005, Extensions quasifrobeniusiennes avec equivalence et dualite de Morita, Afrika Mat., 16(3), pp. 43-54.

Djiby Sow and Mamadou Sanghare

Departement de Mathematiques et Informatique,

Faculte des Sciences et Techniques,

Universite Cheikh Anta Diop, Dakar Senegal

E-mails: sowdjibab@ucad.sn, sowdjibab@yahoo.fr, mamsanghare@ucad.sn
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