A note on prime modules

Di ulgaciones Ma em´a icas Vol. 8 No. 1 (2000), pp. 31–42 A No e on P ime Modules Una No a sob e M´odulos P imos Ch is ian Lomp ([email p o ec ed]) Cen o de Ma em´a ica da Uni e sidade do Po o P a¸ca Gomes Teixei a 4099-002 Po o, Po ugal. Ali io J. Pe˜na P. ([email p o ec ed]) Labo a o io de ´ Algeb a Te´o ica y Compu acional (LATyC) Depa amen o de Ma em´a ica y Compu aci´on Facul ad Expe imen al de Ciencias. La Uni e sidad del Zulia. Apa ado Pos al 526 Ma acaibo 4001. Venezuela. Humbe o Fe n´andez-Mo ´an Villalobos in memo iam Abs ac In his no e we compa e some no ions o p imeness o modules exis ing in he li e a u e. We cha ac e ize he p ime le R-modules such ha he le annihila o o e e y elemen is a ( wo-sided) ideal o R, whe e Ris an associa i e ing wi h uni y, and we p o e ha i Mis such a le R-module hen Mis s ongly p ime. These wo no ions a e s udied by Beachy [B75]. Fu he mo e, i Mis p ojec i e as le R/(0 : M)-module hen Mis B-p ime in he sense o Bican e al. [BJKN]. On he o he hand, i Mis ai h ul hen Mis (s ongly) p ime i and only i Mis s ongly p ime (o an SP-module) in he sense o Handelman-Law ence [HL] i and only i Mis o sion ee and Ris a domain. In pa icula his happens i Ris commu a i e. Key wo ds: bounded ke nel unc o , annihila ing se o a module, p ime module, s ongly p ime module, SP-module, o sion ee module. Resumen En es a no a se compa an algunas nociones de p imi ud pa a m´odu- los exis en es en la li e a u a. Se ca ac e izan los R-m´odulos a izquie da Recei ed: 1999/06/01. Re ised: 2000/02/23. Accep ed: 2000/03/02. MSC (2000): 16D80, 16D20. 32 Ch is ian Lomp, Ali io J. Pe˜na P. p imos pa a los cuales el anulado a izquie da de cada elemen o es un ideal (bil´a e o) de R, donde Res un anillo asocia i o con unidad, y se p ueba que si Mes un al R-m´odulo izquie da, en onces Mes ue - emen e p imo. Es as dos nociones son es udiadas po Beachy [B75]. Adem´as, si Mes p oyec i o como R/(0 : M)-m´odulo a izquie da, en- onces Mes B-p imo en el sen ido de Bican e al. [BJKN]. Po o o lado, si Mes iel, en onces Mes ( ue emen e) p imo si y s´olo si M es ue emem e p imo (o un SP-m´odulo) en el sen ido de Handelman- Law ence [HL] si y s´olo si Mes lib e de o si´on y Res un dominio. En pa icula , es o ocu e si Res conmu a i o. Palab as y ases cla e: un o n´ucleo aco ado, conjun o anulado pa a un m´odulo, m´odulo p imo, m´odulo ue emen e p imo, SP-m´odulo, m´odulo lib e de o si´on. In oduc ion In he ca ego y o le R-modules he e exis se e al no ions o p ime objec s which gene alize he well known no ion o p ime ( wo-sided) ideal o an as- socia i e ing Rwi h uni y. In each case, an ideal Po a ing Ris p ime i and only i R/P is p ime as le R-module. In his pape ou o hese no ions a e conside ed: p ime and s ongly p ime due o Beachy [B75], B- p ime due o Bican e al. [BJKN] and s ongly p ime (o SP-module) due o Handelman-Law ence [HL]. In sec ion 1 we gi e some e minology, no a ion and basic esul s. In sec ion 2 we conside some p imeness condi ions o le R-modules exis ing in he li e a u e and some o i s ela ions a e poin ed ou . Each o hese condi ions gene alizes he well known no ion o p ime ideal in a commu a i e ing. In sec ion 3 we cha ac e ize he p ime le R-modules such ha he le annihila o o e e y elemen is a wo-sided ideal o R(P oposi ion 3.1) and we show some p ope ies o his special ype o le R-modules (P oposi ion 3.2). In pa icula , i Mis such a p ime le R-module which is p ojec i e as le R/(0 : M)-module hen Mis B-p ime (Co olla y 3.1). We conside also he commu a i e case (Co olla y 3.2) and he ai h ul case (Co olla y 3.3). Fu he mo e, we p o e ha i Mis a nonze o le R-module such ha he le annihila o o e e y elemen is a wo-sided ideal o R hen Mis p ime i and only i Mis s ongly p ime (Theo em 3.1). A No e on P ime Modules 33 1 Te minology, no a ion and basic esul s By a ing we shall unde s and an associa i e ing wi h uni 1 6= 0. Rwill always deno e a ing. By an ideal o Rwe shall unde s and a wo-sided ideal o R. By a module we mean a uni a y le R-module. We shall deno e by R−Mod he ca ego y o modules wi h ze o objec (0) whose mo phisms ac on he igh side. I Mis a module and Xis a se hen we shall deno e by MX( espec i ely, M(X)) he di ec p oduc ( espec i ely, di ec sum) o |X| (ca dinal o X) copies o M. We shall use he no a ions RMand N≤M o indica e ha Mis a module and Na submodule o M, espec i ely. Le RM, le X⊆Mand le N≤M. Then he se (N:X) := {a∈R:aX ⊆N}is a le ideal o R. In pa icula , i m∈M hen (0 : m) := ((0) : {m}) is called he le annihila o o mand (0 : M) := ((0) : M) is an ideal o Rcalled he annihila ing ideal o M. Fu he mo e Mis said o be ai h ul i and only i (0 : M) = (0); Mis said o be co ai h ul i and only i he e exis elemen s m1, ..., mn∈Msuch ha Tn i=1(0 : mi) = (0), o equi alen ly, i Rcan be embedded in a ini e di ec sum Mno ncopies o M. I is clea ha e e y co ai h ul module is also ai h ul. The ques ion abou when e e y ai h ul le ideal o Ris co ai h ul was conside ed by Beachy-Blai in [BB]. On he o he hand, e e y ai h ul module is co ai h ul i and only i Rcon ains an essen ial a inian le ideal i and only i Rhas an essen ial and ini ely gene a ed socle (see [B71] and [V]). Following Pe˜na [P94] (see also [Ri]), we shall de ine he annihila ing se o a module M, which will be deno ed by A(M), as A(M) := {a∈R:aRm = (0) o some 0 6=m∈M}. No e ha A((0)) = ∅and i Mis a nonze o module hen (0 : M)⊆A(M) and A(M) = S{(0 : N) : (0) 6=N≤M}is a union o ideals o R. Thus, A(M) is closed unde le and igh mul iplica ion by elemen s o R, bu A(M) is no necessa ily an ideal o R. Recen ly Dauns [D] conside ed o which modules M he se A(M) is an ideal o R(called p imal modules), and p o ed ha M is a p imal module i and only i he comple e la ice o med by all he (le , igh o wo-sided) ideals o Rwhich a e con ained in A(M) has a unique coa om (P oposi ion B in [D]). No e ha i Nis a p ope submodule o a module M hen he annihila ing se A(M/N) is called he adjoin o Nin [D], and i is deno ed by adj(N). Independen ly o he wo k o Dauns, Pe˜na in [P99a] asks abou he modules ha ing ze o annihila ing se s, ob aining he ollowing esul in he language o he ke nel unc o s o Goldman [G]: Theo em 1.1 (Teo ema (3.1) in [P99a]) Le Mbe a nonze o le R-module, le τMbe he ke nel unc o associa ed o 34 Ch is ian Lomp, Ali io J. Pe˜na P. M, and le K(R)be he comple e la ice o med by all he ke nel unc o s o e he ca ego y R−Mod. Then he ollowing condi ions a e equi alen : 1. A(M) = (0); 2. Mis a ai h ul p ime module; 3. E e y nonze o submodule o Mis ai h ul; 4. Mis σ- o sion- ee o e e y p ope bounded ke nel unc o σ; 5. τMis an uppe bound in K(R) o he p ope bounded ke nel unc o s. Le Mand Nbe modules. We say ha Mis cogene a ed by N(o Mis N-cogene a ed) i and only i he e is an exac sequence o modules o ype 0−→ M−→ NX o some index se X. Dually, we say ha Mis gene a ed by N(o Mis N-gene a ed) i and only i he e is an exac sequence o modules o ype N(X)−→ M−→ 0 o some index se X. We shall deno e by Cog(N) he class o modules o med by all he N-cogene a ed modules. I is easy o see ha Cog(N) is a class o modules closed unde aking submodules, isomo phic images and di ec p oduc s. A module Mis ai h ul i and only i e e y p ojec i e module is M-cogene a ed (see [B75]). Now, ollowing Wisbaue [W91], we say ha Mis subgene a ed by N(o Mis N-subgene a ed) i and only i Mis isomo phic o a submodule o an N-gene a ed module. The ca ego y o N-subgene a ed modules is deno ed by σ[N]. In his wo k we shall suppose a ce ain amilia i y wi h he no ions and basic cons uc ions abou ke nel unc o s and i s co esponding il e s (see Goldman [G]). Le Mbe a module and le σbe a ke nel unc o . We say ha Mis σ- o sion ( espec i ely, σ- o sion- ee) i and only i σ(M) = M ( espec i ely, σ(M) = (0)). Fu he mo e, we say ha Mis σ-decisi e i and only i Mis ei he σ- o sion o σ- o sion- ee. This no ion o ela i ely decisi e modules has been conside ed ecen ly in Pe˜na [P99c]. Fo mo e de ails on ing and module heo y, e e o Ande son-Fulle [AF]. 2 Some p imeness condi ions o modules Le Mbe a nonze o module. Following Beachy [B75], we say ha Mis p ime i and only i (0 : M) = (0 : N) o e e y nonze o submodule No M. In Dauns [D] i is poin ed ou ha Mis p ime i and only i (0 : M) = A(M). I is easy o see also ha Mis p ime i and only i he quo ien module R/(0 : M) is cogene a ed by e e y nonze o submodule o M. This condi ion A No e on P ime Modules 35 was s udied by Johnson in [J]. See also [W83]. Recen ly, Pe˜na in [P99c] showed ha Mis p ime i and only i Mis σ-decisi e o e e y bounded ke nel unc o σ. Recall ha a ke nel unc o is said o be bounded i and only i i s co esponding il e is bounded (i.e., con ains a co inal se o ideals o R). This cha ac e iza ion o p ime modules leads us o de ine p ime objec in ca ego ies o G o hendieck (o mo e gene ally, in any ca ego y whe e he e exis bounded localizan subca ego ies) and he no ion o a Φ-p ime module in O e ex ensions ela i e o a subse Φ o he se o addi i e endomo phisms o R(see Lam-Le oy-Ma czuk [LLM]). Now, ollowing Bican e al. [BJKN], we say ha Mis B-p ime i and only i Mis cogene a ed by each o i s nonze o submodules. I is easy o see ha B-p ime implies p ime. In [W83] i is poin ed ou ha Mis B-p ime i and only i L·HomR(M, N)6= (0) o e e y pai L, N o nonze o submodules o M. Finally, ollowing Beachy [B75], we say ha Mis s ongly p ime i and only i o e e y nonze o submodule No Mand m∈M he e exis elemen s n1, . . . , nk∈Nsuch ha Tk i=1(0 : ni)≤(0 : m), o equi alen ly, i Mis subgene a ed by e e y o i s nonze o submodules (see [W83]). I is easy o see ha e e y s ongly p ime module is p ime and ha e e y simple module is s ongly p ime. Thus an s ongly p ime module doesn’ need o be ai h ul. The ings o which he e exis s a ai h ul s ongly p ime module we e conside ed by Desale-Nicholson in [DN]. I is well known ha Mis s ongly p ime i and only i Mis σ-decisi e o e e y ke nel un o σ(P oposi ion 1.2 in [B75]). In he li e a u e on module heo y he e is ano he no ion o s ongly p imeness due o Handelman-Law ence [HL]: a nonze o module Mis an SP- module (‘SP’ by s ongly p ime) i and only i o e e y 0 6=m∈M he e exis elemen s 1, ..., n∈Rsuch ha Tn i=1(0 : im) = (0). I is easy o see ha e e y SP-module is a ai h ul s ongly p ime module. Fu he mo e, i is well known ha Ris an SP-module (and in such a case, Ris said o be ale s ongly p ime ing in [HL]) i and only i he e exis s an SP-module i and only i Ris an s ongly p ime module in he sense o Beachy (P oposi ion 1.3 in [B75]) i and only i he e exis s a co ai ul s ongly p ime module (see [W83]). On he o he hand, in [W83] i is poin ed ou ha Ris a p ime ing i and only i he e exis s a ai h ull (co ai h ul) p ime module. The le s ongly p ime ings o Handelman-Law ence coincide wi h he le absolu ely o sion ee ings (i.e., ings which a e o sion- ee ela i e o e e y p ope ke nel unc o ). These ings we e s udied by Rubin in [Ru] and by Viola-P ioli in [VP75] and [VP73]. Using hese ideas and concep s, i is no di icul o p o e ha a nonze o module Mis an SP-module i and only i M is σ- o sion- ee o e e y p ope (idempo en ) ke nel unc o σ(see [P99a]) i and only i he quo ien module R/(0 : M) is ini ely cogene a ed by e e y 36 Ch is ian Lomp, Ali io J. Pe˜na P. nonze o submodule o M. The las condi ion was poin ed ou by Wisbaue in [W83]. Le Mbe a nonze o module and le us conside he ollowing condi ions: (I)Mis cogene a ed by each o i s nonze o submodules (B-p ime); (II)L·HomR(M, N)6= (0) o e e y pai L, N o nonze o submodules o M; (III) (0 : M) = (0 : N) o e e y nonze o submodule No M(p ime); (IV )R/(0 : M) is cogene a ed o e e y nonze o submodule o M; (V) (0 : M) = A(M); (V I)Mis σ-decisi e o e e y bounded ke nel unc o ; (V II)Mis σ-decisi e o e e y ke nel unc o σ(s ongly p ime); (V III)Mis σ- o sion- ee o e e y p ope ke nel unc o σ(SP-p ime). Then (I)⇔(II)⇒(III)⇔(IV )⇔(V)⇔(V I)⇐(V II)⇐(V III). In his wo k i is shown ha i Mis a nonze o module such ha he le annihila o o e e y elemen o Mis an ideal o R hen he condi ions (III)− (V II) abo e a e equi alen . I Mis p ojec i e as le R/(0 : M)-module hen we can add he condi ions (I) and (II). Fu he mo e, i Mis ai h ul hen we can add he condi ion (V III). In pa icula , his las holds i Ris commu a i e. 3 The main esul s Le Mbe a nonze o module. Recall ha Mis said o be o sion ee i o e e y non ze o-di iso ∈Rand 0 6=m∈Mwe ha e m 6= 0. I Ris a domain, i.e., e e y 0 6= ∈Ris a non ze o-di iso , and Mis o sion ee hen Mis an SP-module and o e e y 0 6=m∈Mwe ha e (0 : m) = (0). No e also ha i m∈Msuch ha (0 : m) is an ideal o R hen (0 : m) = (0 : Rm). Now, Mis called comp essible i and only i Mcan be embedded in any non- ze o submodule o M, and Mis said o be semi-comp essible i and only i M is ini ely cogene a ed by each o i s nonze o submodules (see [BB]). We ha e he ob ious chain o condi ions: comp essible ⇒semi-comp essible ⇒B-p ime. P oposi ion 3.1 Le Mbe a nonze o module, and le S=R/(0 : M). Then he ollowing condi ions a e equi alen : (a) Mis p ime and he le annihila o o e e y elemen o Mis an ideal; (b) (0 : m) = (0 : M) o e e y 06=m∈M; A No e on P ime Modules 37 (c) o e e y 06=m∈M he e exis s an isomo phism o le R-modules :Rm →Ssuch ha (m) =¯ 1; (d) o e e y 06=m∈M he e exis s a homomo phism o le R-modules :Rm →Ssuch ha (m) =¯ 1; (e) e e y non-ze o cyclic R-submodule o Mis isomo phic o RS; ( ) Mis a o sion ee le S-module and Sis a domain. P oo . I is easy o see ha (a)⇔(b)⇔(e) and (c)⇒(d). (b)⇔( ) I Msa is ies (b) hen Shas no ze o-di iso s, because ¯ ¯s=¯ 0 (whe e ¯ = + (0 : M)) implies ¯ ¯sm = sm = 0 o any 0 6=m∈M. Thus, ∈(0 : sm) = (0 : M) o s∈(0 : m) = (0 : M) implies ¯ =¯ 0 o ¯s=¯ 0. Also SMis o sion ee, because i 0 = ¯sm =sm hen s∈(0 : m)=(0:M) and so, ¯s=¯ 0. I Msa is ies ( ) hen ce ainly o all ∈Rand 0 6=m∈Mwe ha e m = 0 ⇒¯ m = 0 ⇒¯ =¯ 0⇒ ∈(0 : M). Hence (0 : m) = (0 : M) o e e y 0 6=m∈M. (a)⇒(c) : Suppose ha he condi ion (a) holds and de ine :Rm →Sby ( m) := ¯ o each ∈R. Then i is easy o see ha is well de ined and is an isomo phism o le R-modules. (d)⇒(b) : Suppose he condi ion (d) holds and le 0 6=m∈M. Then he e exis s a homomo phism o modules :Rm →Ssuch ha (m) =¯ 1. Now, le ∈Rsuch ha m = 0. Then ¯ 0 = ( m) = (m) = ¯ in S. Hence, (0 : m) = (0 : M) and he condi ion (b) holds.  No e ha i Mis a module sa is ying some o he condi ions o P oposi ion 3.1 hen all non-ze o cyclic submodules o Ma e isomo phic o each o he and he na u al p ojec ion S→Rm wi h ¯s7→ sm o e e y 0 6=m∈Mis an isomo phism o le R-modules. Hence he nonze o cyclic submodules o RM and RSsha e he same p ope ies. Recall ha a domain Dis called a le O e domain i o e e y pai o elemen s x, y ∈D he e exis s, ∈Dsuch ha xs =y 6= 0. Recall also ha , ollowing Fleu y [F], a nonze o module Mis said o be hollow i and only i e e y p ope submodule o Mis small in M. P oposi ion 3.2 Le Mbe a p ime le R-module such ha he le annihi- la o s o elemen s o Ma e ideals o Rand le S=R/(0 : M). Then he ollowing hold: 1. Mcon ains a simple submodule ⇔Sis a di ison ing. 2. Mcon ains a uni o m submodule ⇔Sis a le O e domain. 3. Mcon ains a hollow submodule ⇔Sis a local domain. 38 Ch is ian Lomp, Ali io J. Pe˜na P. 4. Mcon ains a nonze o a inian (noe he ian) submodule ⇔Sis a le a inian (noe he ian) ing. P oo . All he obse a ions ollow om he ac ha , by hypo hesis, e e y cyclic R-submodule o Mis isomo phic o RS. So, i Mcon ains a simple submodule hen RSis simple and hence also Sis simple as a le S-module. Thus i is a di ision ing. I Mcon ains a uni o m (cyclic) submodule hen SS is uni o m and by [GW, Lemma 5.15] he esul ollows. The es is simila .  Theo em 3.1 Le Mbe a nonze o le R-module such ha he le annihi- la o s o elemen s o Ma e ideals o R, and le S=R/(0 : M). Then he ollowing hold: 1. RMis p ime i and only i RMis s ongly p ime. 2. I RMis p ime hen SMis an SP-module. 3. RMis B-p ime i and only i RMis p ime and SMis cogene a ed by S as S-module. 4. RMis semi-comp essible i and only i RMis p ime and SMis ini ely cogene a ed by Sas S-module. 5. RMis comp essible i and only i RMis p ime and SMis isomo phic o a le ideal o Sas le S-module. P oo . (1) I is well known ha s ongly p ime implies p ime. Suppose ha Mis p ime and le σbe a p ope ke nel unc o such ha σ(M)6= (0). I 06=m∈σ(M) hen, by hypo hesis, he ideal (0 : M) = (0 : Rm) = (0 : m) is in he co esponding il e o σ. Thus, σ(M) = Mand by P oposi ion 1.2 in [B75], Mis s ongly p ime. (2) Clea by de ini ion and P oposi ion 3.1. (3) Suppose ha Mis p ime as le R-module and cogene a ed by Sas le S-module. Then Mis also cogene a ed by Sas le R-module. Now, by hypo hesis, e e y nonze o cyclic submodule o Mis isomo phic o Sand so, e e y nonze o (cyclic) submodule o Mcogene a es M. The e o e, Mis B- p ime. Con e sely, i Mis B-p ime and e e y nonze o cyclic submodule is isomo phic o S hen Mis cogene a ed by Sas le R-module and also as le S-module. Finally, he p oo o (4) and (5) a e analogous o he p oo o (3).  A No e on P ime Modules 39 Co olla y 3.1 Le Mbe a p ime le R-module such ha he le annihila o s o elemen s o Ma e ideals o R, and le S=R/(0 : M). Then he ollowing hold: 1. I SMis p ojec i e hen RMis B-p ime. 2. I SMis ini ely gene a ed and sel -p ojec i e hen RMis semi-com- p essible. 3. I SMis cyclic and sel -p ojec i e hen RMis comp essible. P oo . All he asse ions ollow om he Theo em abo e since in case SMis p ojec i e, Mis cogene a ed by S. No e also ha o SM ini ely gene a ed and sel -p ojec i e, we ha e ha SMis p ojec i e in he ca ego y σ[SM] o M-subgene a ed S-modules (see [W91, 18.3]). As Sembeds in o SM, we ha e σ[SM] = S-Mod and hence SMis p ojec i e.  To unde line he esul look a he ollowing example: he a ional numbe s Q o m a la p ime Z-module ha is no B-p ime, because HomZ(Q,Z) = (0), and hence Qis no cogene a ed by Z. Co olla y 3.2 Le Rbe a commu a i e ing, le Mbe a nonze o R-module, and le S=R/(0 : M). Then he ollowing condi ions a e equi alen : (a) Mis p ime; (b) Mis s ongly p ime; (c) Sis an in eg al domain. Co olla y 3.3 Le Mbe a ai h ul le R-module such ha he le annihila o o e e y elemen o Mis an ideal o R. Then he ollowing condi ions a e equi alen : (a) Mis p ime; (b) Mis s ongly p ime; (c) Mis an SP-module; (d) Mis o sion ee and Ris a domain. Mo eo e , i Mis p ime hen 1. Mis B-p ime i and only i Mis cogene a ed by R.