LINEARY COMPACT INJECTIVE MODULES AND A THEOREM OF VAMOS For Katy and Peter Carl Faith1 A ring R will denote a commutative associative ring

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 :


Carl Faith1
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 : We call the module (A) the local injective hull of R at M, and its endomorphism ring 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 .
R is Vamosi an (classical in [1]) if every local injective hull of R is linearly compact (in the discrete topology) .
In [1] Vamos proved that every Vamosian ring is SISI, and that every local endomorphism ring of a SISI ring is commutative .We shall prove the converse here, and number of subsidiary results : 1 .Any SISI chain ring is Vamosian, in fact, an almost maximal valuation ring (Theorem 9) .
As a consequence, we prove : (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 EndR E-submodule, i .e . is fully invariant .
When any of these hold, then RM is SISI for every maximal ideal M .
Remark : When this is so, then every local 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 indecomposable injective R-module .' The following are equivalent conditions .
(4) Every cyclic submodule of E is FI .
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) implies that A is commutative .This can be done because E is also essential over M as an R-module, and E = E( ;_1 R ), since M --4-E and E is indecomposable .Furthermore M = annEI 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 C M V a E 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, canonically (e .g .[7], p .81,Prop .

.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 : In our context, this means that annAH = 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 kindly been provided by Professor Vámos, who also supplied the following example (2) of a locally Vamosian ring that is not SISI .(2) Let R be a subdirectly irreducible almost maximal torch ring, i .e .a ring R with : (i) at least two maximal ideals such that RM is an almost maximal valuation ring for each M E max R, (ii) a waist P, where P is a minimal prime and a uniserial module, and such that R/P is an h-local domain [i .e .every nonzero prime is contained in a unique maximal ideal, and every nonzero ideal of R/P is contained in just finitely many maximal ideals .]Such rings exist (see [15]) but can not be SISI since R is not self-injective .(An indecomposable self-injective ring is local .) The next result shows that a local endomorphism ring of a SISI ring is "almost" SISI .(V3) Follows from theorems of Morita [4) and Mueller [3], which imply that a commutative ring A has a Morita duality iff the least injective cogenerator E over A satisfies A = EndAE .By Mueller [3] this is equivalent to requiring that both A and E be l .c .A-modules .

. THEOREM .
The following are equivalent donditions on a ring R (1) R is Vamosian .
(-2) R is SISI and every local endomorphism ring is Vamosian .
(3) R is SISI and every local injective module is injectivendo .
(3) - (1) .Let E be a local injective R-module, and A = End ER .By the proof of ( 2) -( 1), E is the lesast injec tive cogenerator over A (assuming injectivendo) hence l .c .over A by Mueller's theorem, But, since R is SISI, every R-submodule of E is an A-submodule, so E is l .c .over R .
This proves R is Vamosian .

EXAMPLES OF VAMOS RINGS
The following examples of Vámos rings are culled from [1J .
(El) R locally Noetherian (at maximal ideals) that is, R M Noetherian for all maximal ideals M .
(E3) Any commutative ring A with a Morita duality .
In [1] Vámos proved that a ring R (E2) satisfies (E3), provided that R is either not a domain, or R is a complete local in .(For domains, this belongs to Matlis) .In (E3), if A has a duality then the least injective cogenerator is l .c .by Mueller [3] .Then every local injective hull is l .c .

APPLICATION TO FPF RINGS
Commutative FPF rings are studied in [10] (and in articles cited ther , where we raised a question : does the existence of injective module 'E over a commutative ring A = EndAE, imply A is FPF?Followina Corollary 8, we show the answer is no in general.
First consider any Vamosian ring R, and local endomor- e .g., let R be any local Noetherian ring, and A be its completion .We next remark that a domain A is FPF iff A is Prufer, and hence a local domain A is FPF iff A is a chain domain .(=ideals form a chain) .Hewever, a complete local domain need not be a chain ring, in fact :

. PROPOSITION .
If A = EndRE is commutative and E injective over R, then A is a chain ring iff E is a chain module over.R (= uniserial = the lattice of submodules is a chain) .This is a special case, namely (3) and ( 6) of the next theorem .To prove it, we employ the-following so-called double annihilator conditions which hold for a module E quasi-injec tive over a ring .R, -with A = End ER .
If E is ara injective cogenerator in mod-R, then conversely R is a right chaira ring if E is a chain left A-module .
(3) If R is commutative, then any chaira R-module E is a chain A-module (without assuming quasi-injectivity) .
(4) If.A is commutative, then A is a chain ring iff E is a chain A-module .(c) A is a chain ring .
When this is so, then every A or R submodule of E is quasi-injective .4) is an immediate consequence of (dac 2), and the fact E is quasi-injective over A by the proof of Propositon 4 .This implies (5) via (1) .In ( 6), (3) shows that (b) -(a), and (a) p (c) by (4) .Then E is indecomposable, so every R-submodule S of E is quasiinjective by the proof of Proposition 4, whence is an A-module as stated in the proof of Proposition 2 .Thus, (a) (b) .
Moreover, any A-submodule is quasi-injective by Proposition 4 .

2 .
A ring R that is locally a SISI chain ring is Vamos ian (Theorem 10) .Von Neumann regular rings are locally Noetherian rings, and are examples of Vamosian rings ([l]) ; we show that polynomial rings over them are also Vamosian (Theorem 12 and Corollary) .A number of unsolved problems are listed .One of the main ones asks if R Vámos (SISI) is inherited by the polynomi.al ring R[x] .This is unknown even for an almost maximal valuation ring R . 1 .THEOREM .The following are equivalent conditions on a ring R :
ring R is SISI iff every local injective module E = E(R/M) R satisfies any of the equivalent conditions of the proposition .PROOF .This follows since every subdirectly 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 .prove that RM is SISI for every maximal ideal M, hence suppose R is local with maximal ideal M .In this case E = E(R/M) R is an injective cogenerator of mod-R .Now let M = R/I be a subdirectly irreducible factor ring, and let E = E(MR ) be its injective hull taken in E .PROOF OF THEOREM 1 .
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 rational extension of H), then the only ideal in A = Qmax(R) that annihilates it is zero .See, e .g .[71, p .80, 19 .32(b),which implies that ann-annAH = E for any dense ideal H of R .
4 .PROPOSITION .If E is an injective R-module with 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 for any ideal I = annAx for some x E E .PROof .As stated in the proof of Theorem 1, EA is quasi-injective, and A = EndAE .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 a f E A .Since M is an A-submodule, a fM C 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 follows from'its quasi-injectivity as in the proof of Proposition 2 .If f : A/I -> E is an embedding of A-modules, then I = annAx, where x = f(1+I) .Conversely, if I = annAx then there is an embedding A/I ----> E sending A + I -ax G a E A .Note, if R is SISI, then every local endomorphism ring, A = End E(R/M) R , is commutative, and the unique simple A-module W embeds in E and coincides with V .=R/M .Thus, the proposition would imply that A is SISI provided only that E is injective over A .This is not in general true for a SISI ring R .In fact, Vámos singles out a class of rings (called classical in [1J) to rectify this deficiency .We say that .R is a Vámos ring , or Vamosian (formerly classical) provided that every local injective hull is linearly compact (l .c.)over R .We employ the terminology injectivendo to indicate when a module F over R is injective over its endomorphism ring A .An ideal I is co-subdirectly irreducible (co-SDI) if R/I is a subdirectly irreducible ring .ring A at any maximal ideal M is the .completion of RM in the topology generated by the co-SDI ideals of RM , and is a l .c .ring .(V3) Every local injective hull E is injectivendo, and l .c .over its endomorphism ring A, equivalently HomA ( E) induces a Morita duality in mod-A (on the full subcategory of l .c .A-modules) .

( 2 )
=*»(1) .Let E be a local injective module of R, and A = EndRE .Since A is Vamosian, then the injective hull F of its unique simple module W is l .c .over A .But, W C--, E and, in fact, coincides with the unique simple R-mod-üle V embedded in E, hence E C» F in mod-A .This implies that E is l .c .over A .But, by (5) of Theorem l, , every R-submodule of E is an A-submodule of E, hence E is l .c .

( 5 )
If R is a right chain ring, arad A is commutative, then A is a chain ring, and, ¡ .a., E is indecomposable(6) If R and A are commutative, then the f .a.e .:(a) E is a chain A-module .(b)E is a chain R-module .

( 7 )
If R and A are commutative, and R is a chain ring, then 6(a)-(c) hold .PROOF .(1) follows from the (dac 1), since the finitely generated A-submodules of E form a chain, hence the lattice of all A-submodules do too ; (2) follows from the fact that annRannEI = I for any right ideal I when E cogenerates mod-R ; (3) is trivial since every A-submodule is an R-submodule when R is commutative ; ( 8 .COROLLARY .If R is a SISI ring, then a local endo-morphism ring A = End E(R/M) R is a chain ring iff E(R/M) is a chain module .In this case, R M is an almost maximal valuation ring'(=AMVR), and A 'is its completion (and an AMVR) .

5 .
Does MVC imply tnat R[x] is (a) Vamosian, (b) SISI assuming R is l .c.? 6 .If R a SISI ring such that every factor ring is of finite uniform dimension, is R Vamosian?If P is a prime ideal of R[x], and P 0 is the contracted ideal in R, thenR[ x] P ^Rp 0 [x]Pexwhere Eex is the extension of P to R P[x](i .e .-Pex=PRP [x]) .O O PROOF .Pex consists of all g(x)in Rp [x] with 0 coefficients in PRP , and Pex is prime since, in general, 0 for any ring A and prime ideal L of A, we haveA[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) 9 Pex .We can 0 write .h(x)= h 0 (x)/c and g(x) = g0(x)/d with c,d E R\P 0 , and g0 (x), hb (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 Rp [x], 0 and .moreover,g ¢ Pex m so h/g E Rp [x] ex U P 12 .THEOREM .. If R is locally Noetherian, then so is any polynomial ring over R in finitely many variables x1 , . ..,xn .In particular, then R[x1, . ..,xn] is Vamosian .