MP-Dimension of a Meta-Projective Duo-Ring

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1 Applied Mathematical Sciences, Vol. 7, 2013, no. 31, HIKARI Ltd, MP-Dimension of a Meta-Projective Duo-Ring Mohamed Ould Abdelkader Ecole Normale Supérieure de Nouakchott B.P. 990, Nouakchott, Mauritanie alada@univ-nkc.mr Copyright c 2013 Mohamed Ould Abdelkader. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We extend the MP-dimension notion to the meta-projective duorings. In Theorem 5.5 and Theorem 5.7, we make use of this notion to give sufficient conditions on commutative noetherian rings to be artinian semi-simple. Some results on artinian rings are discussed. Mathematics Subject Classification: 13E10, 13C05 Keywords: Duo-ring, meta-projective module, projective module, reflexive module, MP-dimension 1 Introduction Meta-projective modules on a ring R were essentially studied by Feng Lianggui and Tong Wenting in [3] when R is a commutative ring. While Duo-rings were studied by A. L. Fall in [2] and by M. Sanghare in [8]. The results of [3] were extended by M. O. Abdelkader in [5] to the duo-ring case by providing characterizations of Duo-rings. In this note, we give an extension of the MP-dimension notion on a duo-ring, and we study some aspects of wide class of noetherian and artinian rings. 2 Preliminary Notes We start by the notions which will come in force in all what follows. Definition 2.1 Let R be a given ring,

2 1538 Mohamed Ould Abdelkader 1. we say that R is a left duo-ring if every left ideal in R is twosided ideal. 2. R is said to be a right duo-ring if every right ideal in R is twosided ideal. 3. R is a duo-ring if it is both a time left and right duo-ring. The following result gives a characterization of duo-rings. Theorem 2.2 A ring R is a duo-ring if, and only if, Ra = ar, for every a R. Proof: Suppose that R is a duo-ring and pick a R. For every x R, since ar is right ideal then, xa ar. So, there exists y R such that xa = ay. This means that Ra ar. With the same manner, ar Ra. This gives ar = Ra. In other hand, assume this equality is true for every a R and consider a left ideal I in R. For every a I and x R, we have xa Ra = ar I. Theorem 2.3 Each homomorphic image of a duo-ring is a duo-ring. In particular, the quotient of a duo-ring is a duo-ring. Proof: Let R and G be rings, with R a duo-ring and f : R G a homomorphism. Set R = f(r) and y = f(x), with x R. We have yf(r) =f(x)f(r) =f(xr) =f(rx) =f(r)f(x) =f(r)y. By Theorem 2.2, f(r) is a duo-ring. The last assertion derives easily from this. Theorem 2.4 Any product of duo-rings is a duo-ring. Proof: Let (R i ) i a family of duo-rings and R =Π i R i its product. Let us prove that for every x =(x i ) i R, xr = Rx. Let r =(r i ) i R. We have, xr =(x i r i ) i Rx. There exists, for each i an r i R i such that x i r i = r x i. set r =(r i ) i R. Then xr = r x and then xr Rx. So, xr Rx. With the same way, Rx xr and the equality holds. Hence, R =Π i R i is a duo-ring whenever each R i is, by Theorem Meta-projective modules For a given R module M, denote by J M the family of maximal injective quotient left sub-modules of M and by P M that of all maximal quotient projective ones. This leads to the following

3 MP-dimension of a meta-projective duo-ring 1539 Definition 3.1 Let R be a ring and M an R-module. A sub-module N of M is said to be maximal quotient projective sub-module of M, if N is a maximal sub-module and the quotient R-module M/N is projective. We define P (M) to be the intersection of all maximal projective quotient sub-modules of M. That is, P (M) = N. If P M is the empty set, then we write simply N P M P (M) =M. Analogically, we give Definition 3.2 Let R be a ring and M an R-module. A sub-module N of M is said to be a maximal injective quotient sub-module of M, if M/N is injective and simple. We define Q(M) as the intersection of all maximal injective quotient sub-modules of M. That is, Q(M) = N. If J M is the N J M empty set, then we write simply P (M) =M. This allows us to give the following Definition 3.3 An R-module M is said to be meta-projective module if P (M) =0. We say that it is meta-injective module if Q(M) =0. And we have meta- Theorem 3.4 A semi-simple R-module is meta-projective (resp. injective) if, and only if, it is projective (resp. injective). We remember that in [5], it is stated that 1. Let R be a ring and (M i ) i I a family of meta-projective R-modules. Then, i I M i and i I M i are meta-projective R modules. 2. If M is an R module then, the R-modules M,M,M,...are metaprojective. Here, M = Hom(M,A), M = Hom(M,A),... 4 Meta-projective Duo-rings In what follows, we assume that all the module involved in this section are unitary. In the next results we will need the following Theorem 4.1 ([5]) Let R be a duo-ring. The following assertions are equivalent :

4 1540 Mohamed Ould Abdelkader 1. R is a ring with P (R) =0; 2. the semi-reflexive R-modules are meta-projective ; 3. the projective R-modules are meta-projective ; 4. the free R-modules are meta-projective ; 5. the projective R-modules of finite type are meta-projective ; 6. the semi-reflexive R-modules of finite type are meta-projective ; 7. R is isomorphic to a product of projective simple R-modules ; 8. the reflexive R-modules are meta-projective. In the next section we present the main results of this paper. 5 MP-Dimension Let us suppose that R is a meta-projective Duo-ring (i.e P (R) = 0). Note that every projective R-module is meta-projective, so, in which case we can construct a meta-projective resolution for evey R-module. Definition 5.1 Let R be a meta-projective Duo-ring (with P (R) =0) and M an R-module : 1. we define the MP-dimension of M as the smallest natural number n for which there exists an exact sequence 0 P n... M 0, where every P i is meta-projective. We denote this dimension by MP dim R M, or simply MP dimm, if it is clear from the context. 2. With a similar way, we define the MP-dimension of R (noted by MP dimr) as MP dimr = sup {MP dimm, M is an R module}. We introduce also the following Definition 5.2 The global dimension of R, noted by gldimr, is defined as gldimr = sup {MP dims, S is simple R module} Theorem 5.3 Let R be a Duo-ring such that P (R) =0and M be an R- module. Then MP dimm is either 0 or 1. That is, MP dimr {0, 1}.

5 MP-dimension of a meta-projective duo-ring 1541 If P is projective then, the preceding Theorem applies on the following exact sequence : 0 Kerf P f M 0. Now, we study the Duo-rings with the property that MP dim is 0. Such an example is given by a ring which is commutative, artinian and semi-simple. Before to give our main results we give the following lemma due to [3]. Lemma 5.4 Let R be a Duo-ring such that P (R) =0, then the following assertions are equivalent : 1. MP dimr =0; 2. All R-modules are meta-projective ; 3. All R-modules are semi-reflexive. Proof : 1. is equivalent to 2. by the definition of the MP-dimension of R. 2. is equivalent to 3. by using theorem 4.1. Now for the first main result. Theorem 5.5 Let R be a Duo-ring such that P (R) =0and MP dimr = 0. Then, 1. if R is a noetherian Duo-ring, it is semi-simple artinian Duo-ring ; 2. if R is right semi-artinian Duo-ring, then R is also semi-simple artinian Duo-ring. Proof : 1. Since R is noetherian Duo-ring, as the global dimension of R is the upper bound of the numbers MP dims where S runs over the set of simple R modules, and by lemma 5.4 these S s are meta projective simple and then projective, then gldimr = 0. This proves We only need to state the result for the right semi-artinian rings. We have gldimr = sup {P dims/s is un A module simple}. Suppose that R is a meta-projective Duo-ring. In the case of an R-module M, we can apply the definition 5.1 of the MP dim to finish the proof. Remark 5.6 The MP dimr can be considered as measure on metaprojective R modules. The following result is related to split exact sequences.

6 1542 Mohamed Ould Abdelkader Theorem Let M, M 1 and M 2 be R modules and consider the split exact sequence 0 M 1 M M 2 0. a) if M is meta-projective, then MP dimm 1 = MP dimm 2 ; b) if M 1 is meta-projective, then MP dimm = MP dimm Let P, Q and Q be R modules with P projective and 0 P f Q Q 0 be an exact sequence. If n 1 and 0 Q n... Q 1 Q 0 Q 0 is a meta-projective resolution of Q, then 0 Q n Q 2 Q 1 P Q 0 Q 0 is a meta-projective resolution of Q. Proof : (1) Since M = M1 M 2, we have P (M) = P (M 1 ) P (M 2 ) by the Theorem 5.3 which gives the result. (2) Let C : 0 P P 0 be a projective resolution of P and C :0 Q n Q n 1 Q 1 Q 0 Q 0 be a meta-projective resolution. So, there exists a homomorphism of complex F : C C induced by f. Since f is a homomorphism ; the canonical mapping Cokerf has metaprojective resolution : MC(F ) n MC(F ) n 1 MC(F ) 2 MC(F ) 1 MC(F ) 0 Cokerf 0, Hence, MC(F ) n = C n C n 1 and MC(F ) 0 = Q 0. This implies that Q has the meta-projective resolution 0...Q n Q 2 Q 1 P Q 0 Q 0; This finishes the Proof. Obviously, the proof of theorem 5.7 above implies that we can give a simple proof of the well-known result on projective dimension : for an exact sequence then 0 A B C 0, PdC PdA+1, when PdA = PdB or PdB =0. ACKNOWLEDGEMENTS. This work was done while the author was visiting Dakar. He would like to thank l agence universitaire de la francophonie for the financial support. Thanks are also given to Professor M. Sanghare for good guidance and help. References [1] F. W. Anderson and K. R. Fuller : Rings and categories of modules. Springer-Verlag New-York [2] A. L. Fall, Sur les l-duo-anneaux. Thèse 3ème Cycle UCAD, Juin 1999.

7 MP-dimension of a meta-projective duo-ring 1543 [3] Feng L. and Tong W. : On the Ring R with P (R) = 0. Communication in Algebra, 24 (4), (1996). [4] Ould Abdelkader Mohamed : Les anneaux équilibrés. Mémoire of DEA - UCAD, Juillet [5] Ould Abdelkader Mohamed : Meta-projective Duo-rings. JP Journal of Algebra, Number Theory and Applications. To appear. [6] J. Querre : Cours d Algèbre. Masson Paris (1976). (1981). [7] G. Renault : Algèbre non commutative. Gauthier-Villars (1975). [8] M. Sanghare : S-duo-rings Comm. in Algebra 2O(8), (1992). [9] S. D. Toure : Caractérisation des duo-anneaux pour lesquels les modules vérifiant les propriétés (S) sont infiniment générés. Thèse of 3ème Cycle. UCAD, Juillet (2000). Received: October, 2012