Lineary compact injective modules and a theorem of Vamos

Pub . Mat . UAB Vol . 30 n2 2-3 Des . 1986 LINEARY COMPACT INJECTIVE MODULES AND A THEOREM OF VAMOS For Katy and Peter Carl Faith 1 A ring R will denote a commutative associative ring with unit . After Vámos, R is a SISI Ring if every subdirectly irreducible factor ring is self-injective . Let M be a maximal ideal of R and let E = E(R/`4)R denote the injective hull of the simple'R-module R/M, and let A(M) denote the endomorphism ring . Now E is canonically a module over the local ring RM of R at M, and the unique simple RM -module embeds in E canonically . Moreover : E = E ( R/M )R = E(Rri/MRCR)RM We call the module (A) the local injective hull of R at M, and its endomorphism ring A(M)  =End E(R/M) R = End E R-  (B) the-local endomorphism ring of R at M . 1 A part of this paper was written spring semester 1986 at the CRM of Institut d'Estudis Catalans of Barcelona, while I was holding a Rutgers University Faculty Academic Study Program (FASP) . I have the pleasure of thanking Professor Pere Menal for , inviting me to collaborate, Professor Jatmie Moncasi for his many cordial arrangements on my behalf, and other members of the Faculty and staff for helping to make the stay such happy and mathematically profitable one . 127 R is Vamosi an (classical in [1]) if every local injective hull of R is linearly compact (in the discrete topology) . In [1] Vamosproved that every Vamosian ring is SISI, and thateverylocalendomorphism ring of a SISI ring is commutative . We shall prove the converse here, and number of subsidiary results : 1 . Any SISI chainring is Vamosian, in fact, an almost maximalvaluation ring (Theorem 9) . As a consequence, we prove : 2 . A ring R that is locally a SISI chain ring is Vamos ian (Theorem 10) . Von Neumann regularrings are locally Noetherian rings, and are examples of Vamosian rings ([l]) ; we show that polynomialringsover them are also Vamosian (Theorem 12 and Corollary) . A numberof unsolved problems are listed . One of the main ones asks if R Vámos (SISI) is inheritedby the polynomi .a l ring R[x] . This is unknown even for an almostmaximalvaluation ring R . 1 . THEOREM . The following are equivalentconditionson a ring R : (1) R is SISI . (2) Every local endomorphism ring of R is commutative . (3) Every R-submodule of every local injective hull of R is quasi-injective . (4) Every R-submodule of every local injective module E is an End R E-submodule, i .e . is fullyinvariant . When any of thesehold, then RM is SISI for everymaximal ideal M . Remark : When this is so, then everylocal endomorphism ring of R is "almost"SISI ; See Proposition 4 . For the proof, we need a result implicit in [1] . 2 . PROPOSITION . Let E be an indecomposableinjective R-module .  ' The following are equivalent conditions . (1) Every submodule - of E is quasi-injective . (2) Everysubmodule of E is fully invariant (FI) . (3) Everycyclicsubmodule of E is quasi-injective . (4) Everycyclicsubmodule of E is FI . (5) If a cyclicmodule R/I embeds in E then R/I is a self-injective ring . (6)  For each a EA = End ER and x E E,  there exists r E R such that a(x) = xr . When this is so, then A = End RE is commutative . PROOF .  (1) < * - (2)  by a theoremof Johnson and Wong ([7], p .63 . Cor 19 .3), which states that an R-module M is quasi-injective iff M is fully invariant in E(M R ) . If ev ery cyclicsubmodule of  E  is fully-invariant (quasi-injective) then every submodule is, hence (1) - (4) are equivalent . Obviously (4) - (6) . Furthermore (5) - (3), because R/I self-injective implies R/I is quasi-injective (since every R-submodule is an R/I-submodule) . (3) =»(5) . If R/I is quasi-injective qua R-module, it is quasi-injective qua R/I-module, equivalently, self - injective  by Baer's criterion  ([6], p .157, Theorem 3 .41 .) Evidently, (6) impliesthat A is commutative . 3 . COROLLARY . (Vámos) A ring R is SISI iff every local injective module  E = E(R/M) R  satisfies any of the equivalent conditionsof the proposition . PROOF . This follows since everysubdirectly irreducible factor ring  R/I  embeds in  E(R/M) R ,  where  R/M = socle R/I ; and conversely, if  R/I - E(R/M), where  M  is maximal, then R/I is subdirectly irreducible . (1) ° (2) by Vámos [1], and (1) « (3) by Corollary 3 . Moreover in view of (A) and (B), Corollary 3 also yields (3) - (4) . (2)  (1) . It suffices to prove that R M is SISI for every maximal ideal M, hencesuppose R is local with maximal ideal  M .  In this case  E = E(R/M) R  is an  injective cogenerator of mod-R . Nowlet M = R/I be a subdirectly irreducible factor ring, andlet E = E(M R ) be its injective hull taken in E . PROOF OF THEOREM 1 . This can be donebecause E is alsoessentialover M as an R-module, and  E = E( ;_1 R ), since M --4 - E and E is indecomposable . Furthermore M = ann E I D E, and M is essential over M as an R, whence as an R-module, so therefore M = E . This shows that E is a fully invariant submodule of E since obviously where K = annAE . so that aM CM V aE A .  Thus The fact that M ---> E implies that E is a cyclic P .-module (e .g ., see [8], p .15, Prop . 5 .5), whence E ^A in mod-A .- Furthermore, Q = Qmax(R)CA A = A/K ^ End ER E ^A^ Biend El = Q  (C) canonically (e .g . [7], p .81, Prop . 19 .21 .) . Since ER is injective, then it is quasi-injective over Q = Biend ER by Corollary 5 .6A, p .15 of [8) . Using (C) we see that A is self-injective . We also need the fact that A is an injective cogenerator over R (since E is) . Thus ev : ery ideal  H  of  R  is the annihilator of an .A-submodule of E  A , . say H = annIG for an ideal G of A . (See [7], p . 190, Corollary 23 .23 .) But if H is chosen to be a dense ideal of R (= R is a rationalextensionof  H),  then the only ideal in A = Qmax(R) that annihilates it is zero . See, e .g . [71, p .80, 19 .32(b), which impliesthat ann-annAH = E for any denseideal H of R . In our context, this means that ann A H = 0, so G = 0 whence H = R . But, then R= A, since for every q E Q ther exists a dense ideal I of R with qI --~ R . This proves that R is self-injective, and hence R is SISI . Remark : (1) A proof of (2) - (1) has kindlybeenprovidedby Professor Vámos, who also supplied the followingexample (2) of a locallyVamosian ring that is not SISI . Suppose (2) holds but (1) fails . As before, we may assume that R is a local subdirectly irreducible ring embedded in E = E(R) . Since E >,¿ R thereexists xE E\R, and since E is an injective cogenerator, there exist homomorphisms such that PartiallyDifferent Proof E --> E and R : E --~ E a (1)  = x,  Q (1)  = 0,  0(x)  y, 0 . Then 0~ G (x)  =  a (1)  76 Cío (1)  = 0 contradicting commutativity of End RE . (2) Let R be a subdirectly irreduciblealmost maximal torch ring, i .e . a ring R with : (i) at least two maximal ideals such that R M is an almostmaximalvaluationring for each ME max R, (ii) a waist P, where P is a minimalprime and a uniserial module, and such that R/P is an h-local domain [i .e . every nonzero prime is containedin a unique maximal ideal, and everynonzeroidealof R/P is contained in just finitely manymaximal ideals .] Such ringsexist (see [15]) but cannot be SISI since R is not self-injective . (An indecomposableself-injectivering is local .) The next result shows that a localendomorphism ring of a SISI ring is "almost" SISI . 4 . PROPOSITION . If E is an injective R-modulewith commutative endomorphism ring A, any A-submodule of E is quasi-injective, and A modulo any ideal I such that A/I C- . E  is self-injective ; equivalently  A/I  is self injective forany ideal I = ann A x for some x E E . PROof . As stated in the proofof Theorem 1, E A is quasi-injective,and A = End AE .  If S C M are A-submodules of E, and f : S + M an A-map, then by quasi-injectivity of E over A, f is induced by af EA . Since M is an A-submodule,  af MC M,  hence f extends to an endomorphism of . M A . This proves quasi-injectivity of any A-submodule M of E . The self-injectivity of A/I followsfrom'its quasi-injectivity as in the proofof Proposition 2 . If f : A/I -> E is an embedding of A-modules, then I = ann A x, where x = f(1+I) . Conversely, if I = ann Ax then there is an embedding A/I ----> E sending  A + I - ax G a EA . Note, if R is SISI, then every local endomorphism ring, A = End E(R/M) R , is commutative, and the uniquesimple A-module W embeds in E and coincideswith V .= R/M . Thus, the proposition would implythat A is SISI providedonly that E is injectiveover A . This is not in general true for a SISI ring R . In fact, Vámossingles out a classof rings (called classical in [1J) to rectify this deficiency . We say that . R is a Vámos ring , or Vamosian (formerly classical) provided that everylocalinjective hull is linearly compact (l .c .) over R . We employ the terminology injectivendo to indicate when a module F over R is injectiveover its endomorphism ring A . An ideal I is co-subdirectly irreducible (co-SDI) if R/I is a subdirectly irreducible ring . VÁMOS THEOREM [1] . If R is Vamosian then : (V1) R is SISI . (V2) The local endomorphism ring A at any maximal ideal M is the .completion of R M in the topology generated by the co-SDI ideals of R M , and is a l .c . ring . (V3) Everylocalinjective hull E is injectivendo, and l .c . over its endomorphism ring A, equivalently Hom A ( E) induces a Morita duality in mod-A (on the full subcategory of l .c . A-modules) . (V3) Follows from theorems of Morita [4)  and Mueller [3], which implythat a commutative ring A has a Morita duality iffthe least injective cogenerator E over A satisfies A = End A E . By Mueller [3] this is equivalent to requiring that both A and E be l.c . A-modules . 5 . THEOREM . The following are equivalent donditions on a ring R (1) R is Vamosian . (-2) R is SISI and every local endomorphismring is Vamosian . (3) R is SISI and everylocalinjectivemodule is injectivendo . PROOF .  (1) - (2) . By Vamos' theorem, R is SISI, and everylocal endomorphism ring A = End ER has l .c . . injective hull E by (V3) . (2) = *»(1) . Let E be a local injectivemodule of R, and A = End RE . Since A is Vamosian, then the injective hull F of its uniquesimplemodule W is l .c . over A . But, W C--, E  and,  in fact, coincides with the uniquesimple R-mod- 11 LEMMA . If P is a prime ideal of R[x], and P0 is the contracted ideal in R, then R[ x] P ^ Rp 0 [x]Pex where  E ex  is the extension of  P  to  R P [x](i .e .-Pex=PR P [x]) . O  O PROOF .  P ex  consists of all  g(x)  in  Rp [x]  with 0 coefficients in  PR P ,  and  P ex  is prime since,  in general, 0 forany ring A and prime ideal L of A, we have A[x]/L[x] - A/L[x] is a domain . Let f(x) = h(x)/g(x) denote an element of the right side, i .e . Let h(x), g(x) E Rp [x], with g(x) 9P ex . We can 0 write .h(x) =h0 (x)/c  and  g(x) = g0(x)/d with c,d E R\P 0 , and g 0 (x), h b (x) E R[x] . Since c,d ¢ P0, then -cdg0 q! P, hence h(x) =h0(x)/cdg .0(x) E R[x] p . The reverse inclusion is  proved similarly, i .e ., if h, g E R[x], and g ¢ P, then we may view h and g as elements of R p [x], 0 and .moreover, g ¢ P ex m so h/g E R p [x] ex U P 12 . THEOREM . . If R is locally Noetherian, then so is any polynomialringover R in finitelymany variables x 1 ,.. .,x n . In particular, then R[x1, . . .,xn] is Vamosian . PROOF . Since  R  is locally Noetherian, then  R P0 [x] is Noetherian for any primeideal P of R[x], and hence, by Lemma 13, so is the localringat P . 13 . COROLLARY . If R is von Neumannregular,then R[x1, . . .,xn1 is Vamosian . 14 . REMARK . R[x] is then semihereditary, and conversely, if the polynomial ring R[x] over a (not necessarily commutative) ring is semihereditary, then R must be von Neumann regular . (See [12, 13, and 14] . (However~general, von Neumann regular ring R does not imply R[x] semihereditary) . 15 PROPOSITION . If R is Vamosian (resp . SISI), then so is every factor ring . PROOF . If R is SISI, then every factor ring obviously is, so suppose that R -is Vamosian, and I is an ideal, and V a simple R/I module, and E the injectivehullof V in mod-R . It is easy to see that the annihilator E of I in E is the injective hull of V in mod-R/I (cf . the proof of (2) =* (1) of Theorem 1) . It follows that É is l .c . over R/I, since E is l.c . over R, hence R/I is alsoVamosian . is R . 16 . COROLLARY . If R[x] is Vamosian(SISI), then so R[x] is monic if it contains a monic polynomial An ideal I of R[x]  is monic if I contains a monicpólynomial, equivalently, R[xl/I is a finitely generated R-modulé . A ring R is called a Mo nica ring if every co-subdirectlyirreducible ideal of R[x] is monic . An ideal I of R is colocal if R/I is a local ring . Example . Any co-SDI ideal I of a SISI ring is colocal, since R/I is then indecomposable injective, hence has local endomorphisin ring which is isomorphicto R/I . In this example I is also co-PF in the sense that R/I is PF . Thus, R/I has a Morita duality, and hence R/I is Vamos . If P is a subcategory of the category RINGS, then for any ideal H of a ring A, we say that H is a co-P-ideal if A/H E P . In this paperinteraliawe havebeeninterested in subcategories of RINGSconsisting of : irreducible (l .e . uniform) rings, local rings, semilocal rings, semiperfect rings, selfinjective rings, PF-rings, and (locally) Noetherian rings . 17 . Theorem . If R is l .c ., theneverymonicideal I of R[x] is co-semiperfect, i .e . for co-local ideals Ii D I, i =  and t > 1 . Consequently, any monic co-irreducible idealof R[x] is co-local . R[ x1 /I = R[X]/I1x . . . XR[ x] /I t Proof . Since I is monic, then R[x]/I is finitely generated over R, and hence by [1] or [3], is 1 .c . as an R-module . By [161, any l .c . ring is semiperfect, so R[x]/I hasthe stated decomposition . 18 . THEOREM (VAMOS [2]) . If R is a Morita ring (i .e ., has a Morita duality), then so does any algebra A over R that is l.c . over R, in particular, that is a finitely generated R-module . 19 .COROLLARY . If R is a Morita ring, then R[x]/I is a Morita ring for any monicideal I, and hence R[x]/I is self-injective for any monic co-SDI ideal I . Proof . Obvious from the aboye theorem of Vamos and the proof of Theorem 17 . 20 . COROLLARY . If R is a MonicaMorita ring, then R[x] is SISI . Proof . By Corollary 19, R[x]/I is Morita,hence Vamos, and therefore SISI, for every co-SDI ideal . 21 . COROLLARY . If R is a l.c . VamosianMonica ring, then R[x] is SISI . Proof . A ring R is Morita iff R is l.c . and Vamosian,according to Mueller'sTheorem stated earlier so R x  is SISI by Corollary 20 . 22 . PROPOSITION . If R is a l .c . ring, then the Mueller-Vamos conjecture implies that R[x]/I is Morita for any monic ideal I . Proof . Obvious, since R[x]/I is a finitely generated moduleover the Morita ring R . 23 . COROLLARY . If R is a l .c . Monica ring, then MVC implies that R[x] is SISI . Monica? 25 . REMARK . If R is SISI, if I is a co-SDI idealof R[x]  and if I n R is a co-SDI ideal of R, then I can show that I is monic iff I is co-local . This resultwill appear elsewhere . Proof . Clearfrom the proof of Corolary 21 . 24 . QUESTION . Is everyMorita,Vámosian, or SISI ring REFERENCES 1 . Vámos, P ., Classical rings, J . Algebra 34 (1975) 114-129 . 2 . ---------, Rings with duality, Proc . London Math . Soc . 35 (1977) 275-289 . 3 . Mueller, B ., Linear compactness and Morita duality, J . Álgebra 16 (1970) 60-66 . 4 . Morita, K ., Duality for modules and its applications to the theory of rings with minimum condition,Sci . Reports, TokyoKyoiku Daigaku 6 (1959) 83-142 . 5 . Matlis, E ., InjectivemodulesoverNoetherian rings, Pac . J . Math . 83 (1958) 207-19 . 6 . Faith, C ., "Álgebra I : Rings, Modules, and Categories", Springer Verlag,Berlin, Heidelberg, New York 1973, Corrected Reprint 1981 . 7 . ---------, "Álgebra II : Ring Theory",Springer Verlag, 1976 . 8 . ---------, InjectiveModulesoverLevitzki Rings, in "InjectiveModules and Injective Quotient Rings" Lecture Notesin Pure and Applied Algebra, Vol . 72 (1982) 1-69 . 9 . Beck, I ., E-injective modules, J . Algebra 21 (1972) 232-249 . 10 . Faith, C ., Commutative FPF ringsarising as split-null extensions, Proc . Amer . Math . Soc . 90 (1984) 181-5 . 11 . Harada, M ., and Ishii, Y . Osaka J . Math . 9 (1972) 217-223 . 12 . McCarthy, P .J ., The ring of polynomialsover a von Neumann regular rings, Proc . Amer . Math . Soc . 3 9 (1973) 253-4 . 13 . Camillo, V .P ., Semihereditary polynomial rings, Proc . Amer . Math . Soc . 4 5 (1974) 173-4 . 14 . Pillay, P ., On semi-hereditarynon-commutative rings, Proc . Amer . Math . Soc . 78 (1980) 473-4 . 15 . Vámos, P ., The decomposition of finitély generated modules and fractionally self-injective rings, J . LondonMath . Soc . (2), 16 (1977) 209-220 . 16 . Sandomierski, F .L ., Linearlycompact modules and local Morita duality, in "Ring Theory" . Academic Press, New York, 1972 . Rebut el día 12 de betembnede 1986 Rutgers, The State University, New Brunswick, Ni 08903 (USA) and Institut d'Estudis Catalans Centre de Recerca Matemática Bellaterra,Apartat 50 (SPAIN)