n-strongly Ding Projective, Injective and Flat Modules

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1 Internatonal Mathematcal Forum, Vol. 7, 2012, no. 42, n-strongly Dng Projectve, Injectve and Flat Modules Janmn Xng College o Mathematc and Physcs Qngdao Unversty o Scence and Technology Qngdao ,Chna Abstract Some propertes o n strongly Dng projectve, njectve and lat modules are studed and some connectons between n strongly Dng projectve, njectve and lat modules are dscussed. At last, these propertes under change o rngs are consdered. Mathematcs Subject Classcaton: 13D05, 13D02 Keywords: n Strongly Dng projectve module, n Strongly Dng njectve module, n Strongly Dng lat modules and commutatve rng 1 Introducton When s two-sded noetheran, Auslander and Brdger [1] ntroduced the G -dmenson, G -dm (M ) or every ntely generated -module M. Several decades later, Enochs and Jenda [2.3] extended the deas o Auslander and Brdger and ntroduced three homologcal dmensons, called the Gorensten projectve, njectve and lat dmensons. D. Benns and N.Mahdou [4] studed a partcular case o Gorensten projectve, njectve and lat modules, whch they call respectvely, strongly Gorensten projectve, njectve and lat modules. Yang and Lu dscussed n [5] some connectons among strongly Gorensten projectve,njectve and lat modules and they study the propertes under change o rngs. J.Gllespe n [6] ntroduced the denton and basc propertes o Dng njectve, projectve and lat module,whch play the role o brant and cobrant objects over Dng-Chen rngs. The author n [7]studed a partcular case o Dng projectve, njectve and lat modules, whch they call respectvely, strongly Dng projectve, njectve and lat modules.

2 2094 Janmn Xng In ths paper, based on the results mentoned above, we generalzed strongly Dng projectve, njectve and lat modules, whch we call respectvely, n strongly Dng projectve, njectve and lat modules and nvestgate the relaton among them. At last, we study these propertes under change o rngs. The rest o the text s organzed as ollows:some dentons and theorems wll be ntroduced n secton 2; the man results o n strongly Dng projectve, njectve and lat modules are descrbed n secton 3. 2 Prelmnares and Notatons Unless stated otherwse, throughout ths paper all rngs are assocatve wth dentty and all modules are untary modules. Let be a rng. We denote by -Mod (Mod- )the category o let(rght) -modules respectvely. The character module Hom Z ( M, Q Z) s denoted by M. 1 ecall that a rght -module E s a called FP-njectve Ext ( F, E) =0 or all ntely presented modules F. ecently, n [7], a rght -module N s called strongly Dng njectve ( or short SD-njectve) there exsts an exact sequence L I I I L wth I njectve and N = ker and whch remans exact ater applyng Hom ( E, ) or any FP-njectve module E. A rght -module M strongly Dng projectve (or short SD-projectve) there exsts an exact sequence L P P P L wth P projectve and M = ker and whch remans exact ater applyng Hom (, F) or any lat module F. An -module M s sad to be strongly Dng lat ( n -SD-lat or short), there exsts an exact sequence o -modules L F F F L wth F lat and M = ker whch remans exact ater applyng E or any FP-njectve module E. By denton, Dng njectve modules are Gorensten njectve. When s Noetheran, strongly Dng njectve modules are strongly Gorensten njectve modules; When s perect, strongly Dng projectve modules are strongly Gorensten projectve modules. 3. The Man esults Denton 3.1. Let n be a postve nteger. An -module M s sad to be n -strongly Dng projectve ( n -SD-projectve or short), there exsts an exact sequence o -modules 0 M Pn L P1 M 0 wth P projectve or 1 n and whch remans exact ater applyng Hom (, F) or any lat module F. A rght -module N s sad n strongly Dng njectve ( n -SD-njectve or

3 n-strongly Dng projectve, njectve and lat modules 2095 short), there exsts an exact sequence o -modules 0 N I n L I1 N 0 wth I njectve or 1 n and whch remans exact ater applyng Hom ( E, ) or any FP-njectve module E. An -module M s sad to be n -strongly Dng lat ( n -SD-lat or short), there exsts an exact sequence o -modules 0 M Fn L F1 M 0 wth F lat or 1 n and whch remans exact ater applyng E or any FP-njectve module E. Usng the dentons, we mmedately get the ollowng results. Proposton 3.2. (1).I ( M ) I s a amly o n SD-projectve modules, then M s n SD-projectve. (2).I ( N ) I s a amly o n SD-njectve modules, then N s n SG-njectve. The next result gves a characterzaton o the n SD-projectve modules. Proposton 3.3. For any postve nteger n, a -module M s n SD-projectve and only there exst an exact sequence 0 M Pn L P1 M 0 where each P s projectve or 1 n and Ex ( M, F ) =0 or any lat module F and any 1. Proo. I M s n SD-projectve, then there exst an exact sequence 0 M Pn L P1 M 0 such that Hom( -, F ) leaves the sequence exact whenever F s lat. We get Ext ( M, F ) =Ext n ( M, F ) or all 1.By nduce, we get Ext ( M, F ) =0 or all 1 n. Thereore ths mples that Ext ( M, F ) =0 or all 1. Contrary, t s trval. Dually, we get the smlar characterzatons o n SD-njectve and lat modules. The next result gves a characterzaton o the n SD-projectve modules. Proposton 3.5. For any M mod-, the ollowng statements are equvalent: (1)M s n SD-projectve; (2) There exsts an exact sequence: 0 M Pn L P1 M 0 n mod- wth P projectve, such that Ext ( M, F ) or any lat module F and 1 ; (3) There exsts an exact sequence: 0 M Pn L P1 M 0 n mod- wth P projectve, such that Hom( -, F ) leaves the sequence exact whenever F s a module wth nte lat dmenson. Proo. Usng standard arguments, ths ollows mmedately rom the denton o n SD-modules. Dually, we get the smlar characterzaton o the n SD-njectve modules and n SD-lat modules.

4 2096 Janmn Xng Next we prove that the tensor product o an n SD-projectve (lat) rght -module and projectve (lat ) let -module s also n SD- projectve (lat). Proposton 3.6. Let be a commutatve rng and Q a projectve let -module. I M s an n SD-projectve rght -module, then M Q s an n SD-projectve -module. Proo. There s an exact sequence 0 M Pn L P1 M 0 n mod- wth P projectve or 1 n. Then 0 M Q Pn Q L P1 Q M Q 0 s exact and P Q s a projectve -module by [8, Ch. 2, 1 Theorem 3] or 1. Let F be any lat -module. Then Ext ( M Q, F) = Hom ( Q, Ext ( M, F)) = 0 by [9, p. 258, 9.20] or all 1. Hence M Q s an n SD- projectve -module by Proposton 3.3. Proposton 3.7. Let be a commutatve rng. I M s an n SD-projectve -module, then M [x] s an n SD-projectve [x] -module. Proo. There s an exact sequence0 M Pn L P1 M 0 where P s projectve. So 0 M[ x] Pn [ x] L P1 [ x] M[ x] 0 s exact n [x] -Mod and P [x] s a projectve [x] -module. Let Q be any lat (N ) (N ) [x] -module. Then Q[x] [x] Q Q Q. Hence Q [x] s a lat [x] -module, and so Q s a lat -module by [10, Proposton 5.11]. Thus Ext [ x] ( M[ x], Q) Ext ( M, Q) =0 by [9, p. 258, 9.21] or all 1, and hence M [x] s an SD-projectve [x] -module. Proposton 3.8. Let be a commutatve rng and F a lat let -module. I M s an n SD-lat rght -module, then M F s an n SD-lat -module. Proo. There s an exact sequence 0 M Fn L F1 M 0 n -Mod wth F lat.. Then 0 M F Fn F L F1 F M F 0 s exact and F F s lat -modules. Let I be any FP-njectve -module and F be a lat resoluton o I, Then Tor ( M F, I) = H (( M F) F) H ( M ( F F)) Tor ( M, F I) =0 or all 1, Snce F I s an FP-njectve -module. Hence M F s an n SD-lat -module. Theorem 3.9. Let be rght coherent and M s an n SD-lat let -module, then M s an n SD-njectve rght -module. Proo. There exsts an exact sequence 0 M Fn L F1 M 0 n -Mod wth F lat. Then 0 M Fn L F1 M 0 s exact n

5 n-strongly Dng projectve, njectve and lat modules 2097 Mod- and F s njectve. Let I be an FP-njectve rght -module. Then Ext ( I, M ) =Tor ( I, M ) = 0 or all 1, and hence M s an n SD-njectve rght -module. Let be a commutatve rng and S a multplcatvely closed set o. Then S 1 = ( S) / ~ = [ a / s a, s S] s a rng and S 1 M = ( M S) / ~ = [ x / s x M, s S] s an S -module. I P s prme deal o and S = - P. Then we wll denote S 1 M, S by M P, P,respectvely. Proposton Let be a commutatve rng and S a multplcatvely closed set o.i B s an ntely generated n SD-projectve S -module, then B s an n SD -lat -module Proo. There exsts an exact sequence 0 M Pn L P1 M 0 n S -Mod wth P projectve or 1 n. Then P s lat S -module by [10.Theorem 5.18]. Let I be any FP- njectve -module. Then we get Ext 1 ( B, S 1 )=0 or any 1, snce B s an ntely generated n SD S -projectve S -module. Thereore Hom (Ext, S 1 1 ( B S 1 ), S 1 S I ) Tor S ( S 1 I, B ) Tor ( I, B ) S S and Hence Tor ( I, B )= 0 by [11. condton O r ] or all 1. Thereore B s an n SD -lat -module Proposton 3.11 Let be a commutatve rng and S a multplcatvely closed set o, then (1) I s coherent and A s an n SD lat -module, then S A s an n SD -lat -module; (2) I A s an n SD lat -module, then S A s an n SD-lat S -module. Proo. (1) There s an exact sequence 0 M Fn L F1 M 0 n -Mod where F lat or 1 n. Then 0 S A S Fn L S F1 S A 0 s exact and S F s lat S -module or 1 n. Hence S F s lat -module. Let I be any FP- njectve -module. Then S Tor (I, S 1 A ) Tor ( S 1 I, A ) =0 by [10. Prop.5.17] or all > m, snce S I s FP-njectve -module by [12.Theorem 3.21]. Hence S A s an n SD -lat -module; (2)There s an exact sequence 0 M Fn L F1 M 0 n -Mod where F lat or 1 n. Then 0 S A S Fn L S F1 S A 0 s exact and S F s lat S -module or 1 n. Let I be any FP-njectve S -module. Then I be any FP- njectve -module by

6 2098 Janmn Xng S [12.Theorem 3.20]. So Tor (I, S 1 A ) Tor ( I, A ) S 1 =0 or all > m. Hence S A s an n SD- projectve S -module. eerences [1] M.Auslander, M.Brdger, Stable module theory.,mem. Amer.Math. Soc. 94, [2] E.E.Enochs, O.M.G.Jenda, Gorensten njectve and projectve modules, Math.Z. Vol. 220(4), (1995) [3] E.E.Enochs, O.M.G.Jenda, On Gorensten njectve modules, Comm.Algebra.Vol.21(10),(1993) [4] D.Benns, N.Mahdou, Strongly Gorensten projectve, njectve and lat modules,j. Pure Appl. Algebra Vol. 210(2), (2007) [5] X.Yang, Z.Lu, Strongly Gorensten projectve, njectve and lat modules., J. Algebra Vol.320(7), (2008) [6] J.Gllespe, Model Structures on Modules over Dng-Chen ngs, Homology, Homotopy and Applcatons, Vol.12(1), ( 2010) [7] Janmn Xng, Strongly Dng Projectve, Injectve,and Flat Modules, Appled Mechancs and Materals Vols (2011) [8] W.T. Tong: An Introducton to Homologcal Algebra. Hgher Educaton, Bejng, [9] J.J. otman, An Introductons to Homologcal Algebra, Academc Press, New York, [10] M.S.Osborne: Basc Homologcal Algebra, Grad. Texts n Math. Sprnger-Verlag, New York, Berln,2003. [11] A. Shamsuddn, Fnte normalzng extensons, J. Algebra 151 (1992) [12] K..Pnzon, Absolutely pure modules, Thess, Unversty o Kentuchy, eceved: March, 2012