ON RELATIVE CHINESE REMAINDER THEOREM

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1 Bull. Korean ath. Soc. 31 (1994), No. 1, pp ON RELATIVE CHINESE REAINDER THEORE YOUNG SOO PARK AND SEOG-HOON RI Previously T. Porter [3] has given a relative Chinese Remainder Theorem under the hypothesis that given ring R has at least one τ-closed maximal ideal (by his notation ax τ (R) φ). In this short paper we drop his overall hypothesis that ax τ (R) φ and give the proof and some related results with this Theorem. In this paper R will always denote a commutative ring with identity element and all modules will be unitary left R-modules unless otherwise specified. Let τ be a given hereditarty torsion theory for left R-module category R-od. The class of all τ-torsion left R-modules, dented by J is closed under homomorphic images, submodules, direct sums and extensions. And the class of all τ-torsionfree left R-modules, denoted by F, is closed under taking submodules, injective hulls, direct products, and isomorphic copies([2], Proposition 1.7 and 1.10). Notation and terminology concerning (hereditaty) torsion theories on R- od will follow [2]. In particular, if τ is a torsion theory on R-od, then a left R-submodule N of is said to be τ-closed (τ-dense, resp.) submodule of if and only if /N is τ-torsionfree (τ-torsion, resp.). A module is called τ-cocritical if F and /N J for each nonzero submodule N of. A left ideal L of R is τ-critical if R/L is τ-cocritical. Follow Porter [3], we denote ax τ () be the set of all maximal τ-closed submodules of and we say ideals I, J are τ-comaximal if I + J is τ-dense in R. Let I 1,I 2,,I n be ideals of R, they are pairwise τ-comaximal in case I i + I j is τ-dense in R whenever i j. For example, if each I i is a maximal τ-closed ideal of R or each I i is a τ-critical ideal, then these ideals are pairwise τ-comaximal. Received November 9, Revised February 17, This work was partially supported by the Basic Science Research Institute Programs, inistry of Education, 1992, and KOSEF-TGRC

2 Young Soo Park and Seog-Hoon Rim The following Lemma 1 and Theorem 2 can be found in [3], we give the proof of Lemma 1 for the completeness of this paper. LEA 1. (Porter, [3]) Let bealeftr-module, and I, J be τ-comaximal ideals in R, then(i J)/IJis τ-torsion. Proof. If x I J,(I + J)x IJ. Since I + J is τ-dense in R, we have that ann(x + IJ)is τ-dense in R. As xwas arbitrary we find ann((i J)/(IJ)) I +J. Thus we have the desired result. The author can find the following relative Chinese RemainderTheorem in [3]. The version of Porter gave us an impression to study it. THEORE 2(PORTER). Let R be a commutative ring and τ be a torsion theoryon R-od. Suppose that ax τ (R) φ and let {I i i = 1, 2,,n}be a finite family of pairwise τ-comaximal idelas in R. For any left R-module, wehave (1) ( n i=1 I i ) ( n i =1 I i ) is τ-surjective and (2) n i =1 / I i is τ-surjective with kernel n i=1 I i The condition ax τ (R) φ was used by the fact that every member in ax τ (R) is prime ideal in R, which is Albu and Nǎstǎsescu s work [1]. In order to drop the condition ax τ (R) φ, we need a lemma, which is useful in the proof of main Theorem. LEA 3. Let R be a commutative ring and {I i i = 1, 2,,n}be pairwise τ-comaximal idels of R. Letbe any left R-module, then we have the following : (1) I i + j i I j is τ-dense in R for each i = 1, 2,,n (2) I i + ( j i I j ) is τ-dense in for each i = 1, 2,,n. Proof. (1) We prove for the case I 1 + D 1 is τ-dense in R, where D 1 = j 1 I j. For the case n = 1 is clear. Assume that I 1 + k j=2 I j is τ-dense in R. Note that I 1 + k+1 j=2 I j contains (I 1 + k j=2 I j)(i 1 + I k+1 ), which is τ- dense in R, thus I 1 + k+1 j=2 is τ-dense in R i.e., the induction step is proved. Consequently I 1 + n j=2 I j = I 1 +D 1 is τ-dense in R. A similar argument shows that for each i = 1, 2,,n, I i + D i is τ-dense in R, where D i = I 1 I 2 I i 1 I i+1 I n. 94

3 On relative chinese remainder theorem (2) For each i = 1, 2,,n, note that I i + D i = (I i + D i ). /(I i + D i ) can be a left R/(I i + D i )-module by the action (r + I i + D i )(m + (I i + D i )) = rm +(I i + D i ) We regard/(i i + D i ) as a homomorphic image of free R/(I i + D i )- module α (R/(I i + D i )) α,by(1)r/(i i + D i ) is τ-torsion and τ-torsion class is closed under direct sum, we have that I i + D i is τ-dense in. THEORE 4.(RELATIVE CHINESE REAINDER THEORE). Let R beacommutative ring and {I i i = 1, 2,,n} be a finite family of pairwise τ- comaximal ideals in R. For any left R-module, wehave (1) ( n i=1 I i ) ( n i =1 I i ) is τ-surjective and (2) n i =1 / I i is τ-surjective with kernel n i=1 I i Proof. (1) The case n = 1 is trivial. Assume the result holds for any left R-module and all families of pairwise τ-comaximal ideals having fewer than n. Consider {I i i = 1, 2,,n} and we denote by P i = j i I j and D i = j i I j We want to show that I i + P i is τ-dense in R. ByLemma3(1), for each i = 1, 2,,n, I i and D i is τ-comaximal ideals in R.Now apply to Lemma 1, we have that I i +D i I i D i is τ-torsion, so its homomorphic image I i +D i I i +P i is τ-torsion. Consider the following short exact sequence, 0 I i + D i I i + P i R I i + P i R I i + D i 0 By the Lemma 3(1), R/I i + D i is τ-torsion module. And the τ-torsion class is closed under extension, so we have R/(I i + P i ) is τ-torsion, thus I i + P i is τ-dense in R. Now we can apply the Lemma 1, and get ( n k=1 I k ) = I i P i I i P i is an τ-epimorphism. Now by the induction hypothesis, I i P i I i ( D i ) is τ- surjection. Thus ( n k=1 I k) ( n k =1 I k ) is τ-surjection. (2) The case n = 1 is clear. We also assume the result holds for any left R-module and all families of pairwise τ-comaximal ideals having fewer than n. Consider the following short exact sequence 95

4 Young Soo Park and Seog-Hoon Rim 0 ( j i I j ) I i D i I i D i + I i 0 By the Lemma 3(2), /(D i + I i ) is τ-torsion. Thus /(D i I i ) is τ-dense in /D i /I i. Now apply the induction hypothesis D i I i j i I j I i n i=1 I i is τ-surjection. Thus we have the desired result. We examine R-submodules {I i i = 1, 2,,n}of in above lemmas and theorems, and consider the following concept in module theoretic sense. DEFINITION. LetbealeftR-module, a set of leftr-submodules of {N i i = 1, 2,,n} is called τ-coindependent in if (i) each N i is not τ-dense in and (ii) N i + j i N j is τ-dense in for each i = 1, 2,,n. For example, given pairwise τ-commaximal ideals of commutative ring R {I i i = 1, 2,,n}, consider left R-submodules {I i i = 1, 2,,n},then the Lemma 3(2) shows that {I i i = 1, 2,,n}is a set of τ-coindependent in. Properties on τ-coindependent submodules can be found in [4]. In here,we mention only the fact related with Relative Chinese Remainder Theorem. PROPOSITION 5. Let R be a ring with identity (R may not be commutative) and let {N i i = 1, 2,,n}beasetofτ-coindependent R-submodules of. Then we have n i =1 N i is τ-surjectivewith kernel n i=1 N i. Proof. The case n = 1 is clear. We assume for any left R-module and all families of τ-coindependent submodules having less than n. Consider the following short exact sequence ; 0 ( n 1 i =1 N i ) N n n 1 i =1 N i N n n 1 i =1 N 0 i + N n By the τ-coindepency of {N i i = 1, 2,,n}, n 1 i=1 N i +N n is τ-densein. Use the induction hypothesis we have the result. 96

5 On relative chinese remainder theorem COROLLARY 6. If ax τ () is finite, then /J τ () is τ-semisimple τ- artinian, where J τ () is the relative Jacobson radical of. Proof. Since ax τ () is finite, J τ () = n i=1 N i, where N i is τ-critical submodules of. Andtheset{N i i=1,2,,n}forms a τ-coindependent submodulesin, then the relative Chinese Remainder Theorem (Theorem 4) gives an τ-epimorphism J τ () n i =1 N i. Hence J τ is τ-semisimple and τ-artinian as left R-module. () References [1]. Albu. T andnǎstǎsescu. C, Décomposition primaires dans les catégories de Grothendieck commutatives, J. Reine Angew. ath. 280 (1976), and 282. [2]. Golan J.S., Torsion Theories, Pitman onographs and surveys in pure and Applied athematics 29, Longman Scientific and Technical, Horlow (1986). [3]. Porter T., A Relative Jacobson Radical with Applications, Colloquia athematica Soc. János Bolyai, 38. Radical Theory, EGER (Hungary) (1982), [4]. Park Y.S. and Rim S.H., On relative dual Goldie dimensions, preprint (1992). DEPARTENT OF ATHEATICS,KYUNGPOOK NATIONAL UNIVERSITY, TAEGU , KOREA DEPARTENT OF ATHEATICS EDUCATION, KYUNGPOOK NATIONAL UNIVERSITY, TAEGU , KOREA 97