6 Proc. Japan Acad., 43 (1967) [Vol. 43,

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1 6 Proc. Japan Acad., 43 (1967) [Vol. 43, 2. Weak Topologies and Injective Modules By Masao NARITA International Christian University, Mitaka, Tokyo (Comm. by Zyoiti SUETUNA, M,J,A., Jan. 12, 1967) Wu showed in his paper [1] that a characterization of selfinjective rings can be given in terms of weak topologies. The aim of this paper is to generalize this result and give a characterization of in jective modules. Throughout this paper, R will denote a ring (not necessarily commutative) with the identity 1. All R-modules considered will be unitary. We shall show that, if an R-module Q has the property: ann Q = 0, where ann Q denotes the ideal of R consisting of all elements annihilating Q, then a necessary and sufficient condition for the module Q to be injective can be given in terms of weak topologies. In addition to it, we shall show at the end of this paper, that such a simple generalization of the theorem due to Wu is not always available in case ann Q ~ Weak topologies. Let Q be a left R-module. Then HomR (Q, Q) can be regarded as a ring with the identity CQ. A will denote this ring HomR (Q, Q). Let M be a left R-module. Then HomR (M, Q) can be regarded as a left A-module, since we have ~p p e HomR (M, Q) for any cp A and for any p e HomR (M, Q). Now we shall give the module Q a structure of topological space with the discrete topology. In connexion with this topology, we shall give the following definition of B-topology on a module M. (cf. Chase [2]) Definition. Let B be a A-submodule of the left A-module HomR (M, Q). Then the coarsest topology on M such that every element of B is a continuous mapping from M into Q will be called the weak topology on M induced by B or simply the B-topology on M. It is easy to see that all subsets of M of the form f 1 Ker Rz c B, i =1, 2,..., n make a base of neighbourhood system of 0(E M) in the B-topology. It is obvious that the B-topology on M is Hausdorff if and only if, for each non-zero x(x E M), there exists,9(9 E B) such that x Ker /9. According to Wu, we shall say B is separating if the weak topology on M induced by B is Hausdorff. It is evident that HomR (Q, Q)-topology on Q is the discrete

2 No, 1] Weak Topologies and Injective Modules 7 topology since HomR (Q, Q) includes the identity mapping cq. Theorem 1. Let CB HomR (M, Q) be a subset of HomR (M, Q) consisting of all continuous R-homomorphisms from M with the B-topology into Q with the discrete topology. Then CB HomR (M, Q) is a left A-submodule of the module HomR (M, Q). Proof. It is evident that the CB HomR (M, Q) is a left Z-submodule of the Z-module HomR (M, Q). Hence it is sufficient to prove that co o CB HomR (M, Q) C CB HomR (M, Q) for any co E HomR (Q, Q). Let p be an arbitrary element of CB HomR (M, Q), then, by the definition of the B-topology, there exist I2,, (9n, ~2 E B, i =1, 2,, n such that Ker p :D f n -~ Ker $z. Since Ker (c o p) i Ker p, it is easy to see that Ker (co o p) f 2=1 Ker R2, Therefore we have proved that c o p is a continuous mapping, i.e, an element of CB HomR (M, Q). In case Q = R, we can define the anti-isomorphism t from the ring HomR (R, R) onto the ring R itself by t(co) = cp(1), c E HomR (R, R). Let M be a left R-module, then it is easy to verify that HomR (M, R) can be regarded as a right R-module. After these considerations, we have immediately the following corollary to the Theorem 1: Corollary. Let B be a right R-submodule of the module HomR (M, R). Then CB HomR (M, R) is a right R-submodule of the module HomR (M, R). The following theorem is a main theorem of this paper: Theorem 2. Let Q be a finitely generated left R-module. Suppose ann Q =0. Then the following four statements (i) (iv) are equivalent: (i) Q is left R-in jective. (ii) B= GB HomR (M, Q) for any left R-module M (not necessarily finitely generated) and any A-submodule B of the A-module HomR (M, Q). (iii) B = CB HomR (M, Q) for any left R-module M and any separating A-submodule B of the A-module HomR (M, Q). (iv) B = CB HomR (I, Q) for any left ideal I of R and any separating A-submodule B of the A-module HomR (I, Q). Proof. (ii)=.(iii) and (iii)=(iv) are self-evident. (iv)=(i) will be proved as follows: Suppose that Q is not left R-injective. Then there exists a left ideal I of R such that the homomorphism *: HomR (R, Q)-->HomB(I, Q) induced by the canonical injection j: I--~R is not surjective. (See Cartan-Eilenberg, Homological Algebra, p. 8.) Set B=j* (HomR (R, Q)), then it is easy to see that this B is a A-submodule of the A-module HomR (I, Q).

3 $ M, NARITA [vol. 43, Let Q = Rg1 + Rg2 + + Rg3, and let h1, h2,, hs be R-homomorphism from R into Q defined by h2(1) = g2, i =1, 2,, s. It is easy to see that f z_1 Ker j *(h2) = I fl ann Q. Since j *(h2) e B, and since we have assumed that ann Q = 0, we can conclude immediately that the B-topology on I is the discrete topology. Therefore it is obvious that every element of HomR (I, Q) is a continuous mapping from I into Q. Since f is not a surjection, we have immediately the following strict inclusion: B(= j *(HomR (R, Q))) CB HomR (I, Q) ( = HomR (I, Q)). Thus we have proved (iv)=(i). The proof of (i)=(ii) is quite similar to the proof which is given by wu in his paper [1]. Suppose that Q is left R-injective, and let M be a left R-module. Let B be a A-submodule of the module HomR (M, Q), and let p be an arbitrary element of CB HomR (M, Q). In order to complete the proof, we have only to show that p is contained in B. Since we have assumed that p is a continuous mapping, there exist 131, 82,.., /3, f9 2 E B, i =1, 2,.., n such that Ker p f ti 1 Ker 132. Let U= Q 0 Q... (+~ Q be a direct sum of n copies of the module Q, and let f : M--~ U be the R-homomorphism defined by f (x) = (31(x), /3(x),, $n(x)) for x E M. Then it is evident that Ker f = f Ker $2. Now let j : Im f--; U be the canonical injection, and let g : M---~Im f be the R-homomorphism such that j o g =f. Since Ker g c Ker p, and since g is an epimorphism, it is easy to see that there exists an R-homomorphism a: Im f--~q such that a o g = p. From the assumption that Q is left R-injective, it is easily seen that the homomorphism 6 can be extended to an R-homomorphism z: U- >Q such that z o f = p. Let z2: Q---Q be the i-th component of z: U-->Q. Then we have z(y) = ~i=1 z2(y) for y e U. Using above results, we have immediately the following equation for any x e M: p(x) = z f (x) =~ 1 z2(~2(x)) This equation shows that p = z2 R2 Since we have assumed that B is a A-submodule of the module HomR (M, Q), and since z2 e A, i =1, 2,.., n, it is obvious that p E B. Thus we have proved (i)=(ii). 2, The case ann Q ~ o. In case ann Q ~ 0, the conclusion of the theorem 2 above stated does not always hold. To make clear this situation, we shall give an example of a ring R and an R- module Q with the following properties: (a) ann Q ~ 0, (b) Q is not R-injective, (c) B = CB HomR (M, Q) for any R-module M and for any A-submodule B of the module HomR (M, Q). Example. Let R be a primary local ring with the maximal

No. 1] Weak Topologies and Injective Modules 9 ideal m = (u), where u2 = 0. (a) Then it is evident that ann m = m ~ 0. (b) It is also easy to see that m is not injective as an R-module.

4 No. 1] Weak Topologies and Injective Modules 9 ideal m = (u), where u2 = 0. (a) Then it is evident that ann m = m ~ 0. (b) It is also easy to see that m is not injective as an R-module. (Actually it is easy to see that the essential injective envelope of the R-module m is R itself,) (c) Hereafter the ideal m will be understood as a topological space with the discrete topology. Let F= R/ m be the residue field, and f : R-->F the canonical epimorphism. Let M be an R-module (not necessarily finitely generated). Since ann m=m, it is easy to see that HomR (M, m) can be regarded as an F -module (i,e, a vector space over F of finite or infinite dimension). It is also obvious that the ring HomR (m, m) is isomorphic to F. Let B be an F submodule of the F -module HomR (M, m). Let CB HomR (M, m) be the set consisting of all continuous homomorphisms from the module M with the B-topology into m with the discrete topology. Let p be an arbitrary element of CB HomR (M, m). For the present purpose, it will be sufficient to prove that p e B..., By the definition of the B-topology, therexist $i, $2, e B, i =1, 2,.., n such that Ker p II9 f z 1 Ker $i Since HomR (M, m) is a vector space, we can choose elements $1,, from $i, $2,, $ in such a way that these $ik are linearly independent over F and any of $l, R2,.., $, can be expressed as a linear combination of Then it is easy to see that f z i Ker $i= f t=1 Ker $it. Therefore it is evident that we can assume without any loss of generality that $i, R2,.., are linearly independent over F. Let 9i(aµ) = ai~u, where {a~}µe~ be an R-basis (not necessarily minimal) of the R-module M. Let a'= f (ail), where f is the canonical epimorphism from the ring R onto the field F, Let W be the linear subspace spanned by all column vectors of the type Then it is self-evident that dim W < n, and it is easy to see that dim W is just equal to n. Now let b1, b2,.., bn be elements chosen from {a/ }µ r In such a way that det (bi,) ~ 0 where Ri(bs) bi, = f (big). Let () be the inverse matrix of the matrix (be), and ci; be a representative of each of ci; such that f (c5) =, 2 =1, 2,, n, j=1, 2,..., ii. Let c = En1 ckbk, then we have $(c)=1 ~n ckr ~ -' k= j i ~ k= j i (bk)=, Ckabiku From the definition of ckj, we have = Therefore we have immediately Ri(ch) = l ju since u2 = 0. Let dz=at-~j=1 ajzc;, z e P. Then it is easy to see that c1, c2,

5 10 M, NARITA [Vol. 43,, cn and {d} z z1 generate the R-module M. From the definition of {d} 1, TTwe have immediately R2(dz) _ i(a) - = au -- a2zu =0. This implies that dr E f Ker j2 c Ker p. Let p(c2) = h2u, i =1, 2, ', n, and let a = p - ~~=1 h3i9. Then we have a(c2) = p(c2) - ~; ^1 h;,~; (c2) =h2u -h2u = 0, i=1, 2,..., n. On the other hand, it is obvious that a (d) = 0, z E r, since dt E Ker p and dz E Ker i =1, 2, ', n. Since c1, c2,, cn and {d}6 ztmake an R-basis of the R-module M, we have immediately a = 0, i.e, p =~~=1 h1/92. Thus we have proved that p belongs to B. References [1] L. E. T, wu: A characterization of self-injective rings. Mathematics, 10, (1966). S. U. Chase: Function topologies on Abelian groups. Mathematics, 7, (1963), Illinois Illinois Journal Journal of of

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