Internatonal Journal of Algebra, Vol. 8, 2014, no. 5, 229-238 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/ja.2014.4212 On P-Duo odules Inaam ohammed Al Had Department of athematcs College of Educaton for pure scence/ Ibn-Al-Hatham Unversty of Baghdad, Baghdad, Iraq Rana Noor ajeed Department of athematcs College of Educaton for pure scence/ Ibn-Al-Hatham Unversty of Baghdad, Baghdad, Iraq Copyrght 2014 Inaam ohammed Al Had and Rana Noor ajeed. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract In ths paper we ntroduce the concept P-dou module as a generalzaton of dou modules, where an R module s called a P-duo module f every pure submodule of s a fully nvarant. any results about ths concept are gven. Also some relatonshps between a P-dou module and other modules related wth t are presented. Keywords: P-duo modules, duo modules, weak duo modules, pure submodules. Introducton Throughout ths paper all rngs are commutatve rngs have an dentty and all modules are unttal. R s a rng and a left R-module. A submodule N of s called fully nvarant f f(n) N for every R-endomorphsm f of. Clearly 0 and are fully nvarant submodules of. The R-module s called a duo module f every submodule of s a fully nvarant. The rng R s a duo rng f t s duo as R-module. It s clearly that every rng s a duo rng. Harmanc and Smth n [1] ntroduced and studed weak duo modules, where an R-module s called
230 Inaam ohammed Al Had and Rana Noor ajeed a weak duo module provded every drect summand submodule of s fully nvarant. In ths paper we ntroduce purely duo module (brefly P-duo module) where an R- module s called a P-duo f every pure submodule of s fully nvarant where a submodule N of s called pure f I N = IN for each deal I of R, [2]. Equvalently, N s a pure submodule f I N = IN for each fntely generated deal I of R. It s well-known that every drect summand of a module s pure. Hence every P-duo module s weak duo. In ths paper we study P-duo module, and we gve some condtons under whch P-duo and weak duo are equvalent. Also we gve some relatonshps between P-duo module and other related modules. Defnton (1): An R-module s a P-duo module f every pure submodule of s a fully nvarant. Remarks and Examples (2): (1) It s clear that every duo module s P-duo and every P-duo s weakly duo, but the converse are not true n generals, for example: The Z-module Q s not duo, for the submodule Z of Q s not fully nvarant, x snce there exsts f:q Q defned by f(x) =, x Q and hence 2 1 f(z) = z Z 2. On the other hand, f s a dvsble module over a P.I.D R, then by [3,Cor.2.9, p.62] every pure submodule of s a drect summand. But Q has only two drect summands namely (0) and Q. Hence Q s a P-duo and a weakduo Z-module. Note that we clam that a weak-duo module need not be a P-duo module, but we have no example. (2) The Z-module Z 2 Z 2 s not weak-duo, snce there exsts f: Z 2 Z 2 Z 2 Z 2 defned by f(x, y) = (y, x), x, y Z2. So f N = Z 2 (0) (whch s a drect summand of Z 2 Z 2 ), then f(n) = (0) Z2 N. Also, Z 2 Z 2 s not a P-duo module and not a duo module. (3) The drect sum of P-duo modules may not be P-duo (see example two). (4) Every multplcaton module s a duo module, hence a P-duo module and a weakly duo module. (5) Every pure smple module (module wth only two pure submodules (0) and s a P-duo module, hence a weak-duo module. (6) A submodule of a P-duo module may not be a P-duo module as the followng example shows: Consder the vector space R 2 over R and let R=R R 2. Defne on R by:
On P-duo modules 231 (a,u) (b,v) = (ab,av + bu) for each a, b R and for each u, v R 2. Then R s a commutatve and so t s a duo R-module and so t s a P-duo module. However the submodule L=(0) R 2 whch s somorphc to R 2 s not a P-duo module, snce the submodule N= {(0,x):x R} s a pure submodule but t s not a fully nvarant submodule. Proposton (3): A drect summand of P-duo module s a P-duo module. Let N be a drect summand of a P-duo R-module. Then = N W for some W. Let K be a pure submodule of N and let f:n N be an R- homomorphsm module. Snce N s a drect summand, then N s a pure submodule n, hence K s a pure submodule n. f(x) x N Defne g: by: g(x) = 0 otherwse g s a well-defned R-homomorphsm. It follows that g(k) K, snce s a P- duo module and K s a pure submodule n. But g(k) = f(k). Hence f(k) K; that s K s a fully nvarant submodule of N. Thus N s a P-duo module. Recall that an R-module s called purely quas-njectve f for each pure submodule N of and for each f:n, there exsts an R-homomorphsm g: such that g = f where s the ncluson mappng [4]. f N g An R-module s called quas-projectve f for each R-module A and every epmorphsm g: and every R-homomorphsm f: N, there s f End() such that g f = f, [5] Proposton (4): Let be a P-duo R-module. Then (1) If s a purely quas-njectve, then every pure submodule of s a P-duo module. (2) If s a quas-projectve, then for any pure submodule K of, K s a P-duo R-module.
232 Inaam ohammed Al Had and Rana Noor ajeed (1) Let L be a pure submodule of N. Let f:n N. Snce N s a pure submodule n and s a purely quas-njectve module, there exsts g: such that g = f where s the ncluson mappng. Thus g (L) = g(l). But L s a pure submodule n N and N s a pure submodule n, mples L s a pure submodule n. Hence g(l) L. Also g (L) = f(l) = f(l). Thus g(l) = f(l) and so f(l) L. Therefore N s a P- duo module. f N N g (2) Let H K be a pure submodule of K and let g: be an R- K K homomorphsm. Let π : be the natural epmorphsm. Snce K K s quas-projectve, there exsts g*: such that g* = g. Hence g*(m) + K = g(m + K) for each m. But H K s a pure submodule n and K s a pure submodule n, K so that H s a pure submodule n. It follows that g*(h) H, snce s a P-duo module. H H Hence g ( ) g *(H) + K. Thus H s a P-duo module. K K K Proposton (5): Let be an R-module such that every cyclc submodule s pure. Then s a P-duo module f and only f for each f End() and for each m, there exsts r R such that f(m) = rm. ) Let f End(), m. Snce <m> s pure, then f(<m>) <m>. Hence the result s obtaned. ) The stated condton mples f(n) N for every f End(). It follows that s a duo module. Hence t s a P-duo module. Proposton (6): Let a module = 1 2 be a drect sum of submodules 1, 2 such that s a P-duo module. Then Hom( 1, 2 ) = 0.
On P-duo modules 233 Snce 1 s a drect sum of, 1 s a pure submodule n. But s a P- duo module, so 1 s a fully nvarant submodule n. Hence Hom( 1, 2 ) = 0 [1,lemma 1.9]. Theorem (7): Let an R-module = 1 2 be a drect sum of submodules 1, 2 such that ann 1 + ann 2 = R. Then s a P-duo module f and only f 1 and 2 are P-duo modules and Hom(, j ) = 0 for j,, j {1,2}. ) It follows by proposton 3 and proposton 6. ) Let N be a pure submodule of. Snce ann 1 + ann 2 =R, then by [6,Theorem 4.2] N = N 1 N 2 for some N 1 1, N 2 2. Hence N 1 s a pure submodule n 1 and N 2 s a pure submodule n 2. Let f: be an R-homomorphsm. Then f j f j : j j, j = 1,2, where j s the canoncal projecton and j s the ncluson map. Hence j f j (N j ) N j, j = 1,2, snce j (j = 1,2) s a P-duo module. oreover by hypothess k f j (N j ) = 0 for k j (k,j {1,2}). Then f(n) = f(n 1 ) + f(n 2 ) 2 ( f )(N ) 2 N N. Thus s a P-duo module. The followng s a well-known lemma. Lemma (8): Let be an R-module such that submodule of, then Theorem (9): Let a module N = (N ). I I j j j j j=1 j=1 =,. If N s a fully nvarant I = of submodules ( I). Then s a P-duo module f and only f (1) s a P-duo module for all I. (2) Hom(, j ) = 0 for all j,, j I. (3) N = (N ) for each pure submodule N of. I ) It follows by proposton 3, proposton 6 and lemma 8. ) Let N be a pure submodule of. By (3), N = (N ). Thus N s a pure submodule n. Let f:. For any j I. Consder the followng I j f j j j
234 Inaam ohammed Al Had and Rana Noor ajeed where j s the ncluson and j s the canoncal projecton. Hence j f j : j j and so j f j (N j ) N j for each j I. By condton (2), Hom(, j ) = 0 j. Hence f( (N )) ( j f j )(N j ) (N j I j ) = N. Thus s a P-duo module. j j I j I Recall that an R-module has the pure ntersecton property (brefly PIP) f the ntersecton of any two pure submodules s agan pure, [3, defnton 2.1,p.33]. Corollary (10): Let = wth for each I. Then s a P-duo module f the I followng condtons hold: (1) s P-duo for every fnte subset I of I. I (2) satsfes PIP. By (1), s a P-duo module for every I. Also j s a P-duo module for each j,, j I. Let N be any pure submodule of. Let x N, hence x = L, for some fnte subset I of I. Thus x N L. By condton (2), N L I s a pure submodule n. But N L L, so N L s a pure submodule n L. Snce L s a P-duo module by (1), N L s a fully nvarant submodule n L. Thus N L = [(N L) ] (N ). It follows that I I I x (N ). Thus N = (N ) I I theorem 9. x (N ) and so and hence s a P-duo module, by In [1], weak duo module satsfes summand sum property and summand ntersecton property, where A module s sad to satsfy summand sum property f K + L s a drect summand of whenever K and L are drect summands of. Also, satsfes the summand ntersecton property f K L s a drect summand of whenever K and L are drect summands of. We prove the followng: Proposton (11): Let be a P-duo module. If K s a drect summand of and L s a pure submodule of, then K L s a pure submodule of.
On P-duo modules 235 Snce K s a drect summand of, = K H for some H. Snce s a P-duo module and L s a pure submodule, then L s a fully nvarant. Hence L = (L K) (L H). Thus L K s a drect summand of L, so L K s a pure submodule n L. But L s a pure submodule n, hence L K s a pure submodule n. Proposton (12): Let be an R-module. Then satsfes (*), f and only f s a P-duo module, where satsfes (*) f for each pure submodules wth zero ntersecton, then ther sum s a fully nvarant submodule n. ) Let N be a pure submodule of. Let H = (0), then H s a pure submodule n and N H = (0). Hence by (*), N = N + H s a fully nvarant. Thus s a P-duo module. ) It s clear. To gve the next result, we gve the followng lemma: Lemma (13): Let be an R-module such that for each deal I of R and for each pure submodules N, W of such that I(N W) = IN IW. Then satsfes PIP. Let N and W be two pure submodules of. Then I N = IN and I W = IW. Hence I (N W) = (I N) W = IN W, also I (N W) = (I W) N = IW N. Hence IN W = IW N. On the other hand, I(N W) = IN IW. We clam that IN W = IN IW. Let x IN W = IW N. Hence x IN IW so IN W IN IW. The reverse ncluson s clear thus IN IW = IN W. Thus I (N W) = IN W = IN IW = I(N W). Therefore s satsfes PIP. Corollary (14): Let =, wth, I. Then s a P-duo module f the I followng condtons hold: (1) s a P-duo module for every fnte subset I of I. I (2) satsfes PIP. It follows drectly by corollary (10) and lemma (11). The followng remark s clear.
236 Inaam ohammed Al Had and Rana Noor ajeed Remark (15): Let be a semsmple R-module. Then the followng statements are equvalent: (1) s a duo-module. (2) s a P-duo module. (3) s a weak duo module. Recall that an R-module s called a fully stable module f every submodule N of s a stable submodule of, where N s called a stable submodule f f(n) N for any R-homomorphsm f:n,[6]. Note that every stable submodule s a fully nvarant submodule. Hence we get the followng result. Corollary (16): Let be a semsmple R-module. Then the followng statements are equvalent: (1) s a P-duo module. (2) s a duo module. (3) s a weak duo module. (4) s a fully stable module. Recall that an R-module s sad to be extendng (or CS), f every submodule of s essental n a drect summand of, [9]. Or equvalently, An R-module s extendng f and only f every closed submodule s a drect summand, [10,proposton 2.4, p.20]. Recall that an R-module s called a purely extendng module f every submodule of s essental n a pure submodule of [8]. It s clear every extendng module s a purely extendng module. A submodule N of an R-module s called closed f t has no proper essental extenson, [11]. Proposton (17): Let be a purely extendng R-module. Then the followng statements are equvalent: (1) s a P-duo module. (2) Every closed submodule of s a fully nvarant. (3) s a weak duo module. (1) (2) Let N be a closed submodule of. Snce s a purely extendng module, N s essental n a pure submodule (say H). Hence N = H, that s N s a pure submodule. It follows that N s a fully nvarant, snc s a P-duo module. (2) (3) Let N be a drect summand of. Hence N s a closed submodule of [11, exercse3, p.19] and so that N s a fully nvarant submodule of.
On P-duo modules 237 Corollary (18): Let be an extendng module over a PIR. Then the followng statements are equvalent: (1) s a P-duo module, (2) Every closed submodule of s a fully nvarant submodule of. (3) s a weak-duo module. (1) (2) (3) It follows by proposton 15. (3) (1) Let N be a pure submodule of. Snce R s a P.I.R then by [7,excersce 15,p.242], N s closed, and snce s extendng N s a drect summand and so by (3) N s a fully nvarant submodule of. Thus s a P-duo module. Recall that an R-module s sad to be regular f every submodule of s pure [12]. Proposton (19): Let be a regular R-module. Then the followng statements are equvalent: (1) s a P-duo module, (2) s a duo-module, (3) s a weak duo module. (1) (2) It s clear, snce every submodule of s pure. (2) (1) and (2) (3) are obvous. (3) (2) Let N. Snce s regular and s a weak duo. Hence by [13,proposton 2.2.10,p.44] s a fully stable, so every submodule of s stable. But every stable submodule s a fully nvarant. Thus s a duo module. It s well-known that every drect summand s pure but not conversely. Also t s clear that when every pure submodule s a drect summand, then P-duo module and weak duo module are equvalent. However under certan classes of modules, pure submodule s a drect summand, as the followng result, [3]. Lemma (20): Let be an R-module. Then (1) If s a prme njectve R-module, then every puure submodule s a drect summand [3,proposton 2.7,p.(61-62)]. (2) If s a dvsble module over a P.I.D., then every pure submodule s a drect summand [3,corollary 2.9,p.62]. (3) If s a fntely generated module over a Noetheran rng, then every pure submodule of s a drect summand [3,proposton 2.10,p.62]. (4) If s a Noetheran projectve R-module, then every pure submodule s a drect summand [3,proposton 2.11,p.63].
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