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On Subrings of Rings 1 Introduction International Journal of Algebra, Vol. 6, 2012, no. 25, 1233 - 1236 On Subrings of Rings Yıldız Aydın Ondokuz Mayıs University Faculty of Art and Science, Department of Mathematics Kurupelit, Samsun, Turkey [email protected] Ali Pancar Ondokuz Mayıs University Faculty of Art and Science, Department of Mathematics Kurupelit, Samsun, Turkey [email protected] Abstract We characterize HR-max subring which is to be inspired by the notion of c-normal subgroup where c-normal subgroup introduced by Wang. Also we obtain some basic properties of HR-max subrings. Mathematics Subject Classification: 13A15 Keywords: Subring, ideal, maximal ideal, HR-max subring 1 Introduction Subrings and maximal ideals of rings play an important role to studying rings. We determined the subrings which contains a maximal ideal. The concept of c-normal subgroup introduced by Wang in [8]. A subgroup H of a group G is said to be c-normal subgroup if there exists a normal subgroup N such that HN = G and H ∩ N ≤ HG = CoreG(H). As a correspondence of this concept in ring theory we called a subring H of a ring R is HR-max subring if there exists an ideal N of R such that R = H + N and H ∩ N ≤ HR, where HR is maximal ideal of R contained in H. We obtained some properties in an associated ring R with such subrings. Firstly every maximal ideal of a ring R is also HR-max subring. If H is a HR-max subring satisfying H ≤ K ≤ R, where K is a local subring of R, then H is a HK -max subring of K and finally the class of HR-max subrings of a ring R is closed under formation of images. 1234 Y. Aydın and A. Pancar 2 Preliminary Notes Definition 2.1 A ring R is called duo if every left ideal of R is ideal of R. Therefore if R is a duo ring, every ideal of R is characteristic ideal, namely, if I is an ideal of R and R = K ⊕ L then I =(I ∩ K) ⊕ (I ∩ L) [6]. The following lemmas are crucial for our study. For the proofs see [7]. Lemma 2.2 Let R be a ring, A, B, C be subrings of R and B ≤ C. Then (A + B) ∩ C =(A ∩ C)+B. Lemma 2.3 Let R be a ring, A be a subring of R and B be an ideal of R. Then A ∩ B is an ideal of A. Lemma 2.4 Let R be a ring, H be a subring of R such that K ≤ H ≤ R and K is an ideal satisfying K ≤ N ≤ R where N is an ideal of R. Then R = H + N if only if R/K =(H/K)+(N/K). 3 Results Definition 3.1 A subring H of a ring R is said to be HR-max subring if there exists an ideal N of R such that R = H + N and H ∩ N ≤ HR, where HR is maximal ideal of R contained in H. Theorem 3.2 Let R be a commutative artinian ring. Then R is direct sum of finitely many HR-max subrings. Proof: Since R is a commutative artinian ring, then R is direct sum of finitely many local artinian ideals [1]. Therefore R has local artinian ideals I1,I2, ..., In such that R = I1 ⊕ I2 ⊕ ... ⊕ In. Since the ideals I1,I2, ..., In are local, 1 ≤ i ≤ nIi has a unique maximal ideal, say Mi. Now for each 1 ≤ i ≤ n, R = Ii +(I1 ⊕ I2 ⊕ ... ⊕ Ii−1 ⊕ Ii+1 ⊕ ... ⊕ In) and Ii ∩ (⊕i=jIj)=0≤ Mi ≤ Ii. Hence the ideals I1,I2, ..., In are HR-max subrings and R is direct sum of finitely many HR-max subrings. Theorem 3.3 Let R be a ring and H be an arbitrary proper ideal of R. Then H is a HR-max subring of R if and only if H is maximal ideal of R. Proof: Let H be HR-max subring of R. Then there exists an ideal N of R such that R = H + N and H ∩ N ≤ HR ≤ H. Since HR ≤ H<Rand HR is a maximal ideal of R, HR = H. Hence H is maximal ideal of R. Conversely let H be maximal ideal of R then HR = H and since H is maximal ideal of R there exists xR\H such that R = H + Rx and H ∩ Rx ≤ H = HR. Therefore H is HR-max subring of R. On subrings of rings 1235 Theorem 3.4 Let R be a ring, H be a subring of R and K be a local ideal of R.IfH is a HR-max subring of R satisfying H ≤ K ≤ R, then H is a HK-max subring of K. Proof: Let H be a HR-max subring of R and H ≤ K ≤ R. Then there exists an ideal N of R such that R = H + N and H ∩ N ≤ HR. By Lemma 2.2, K = K ∩ R = K ∩ (H + N)=H +(K ∩ N), and K ∩ N is an ideal of K and H ∩ (K ∩ N)=(H ∩ N) ∩ K ≤ HR ∩ K, where by HR ∩ K is ideal of K by Lemma 2.3. Now, HR ∩ K ≤ HR ≤ H ≤ K ≤ R, and HR ∩ K is an ideal of R contained in H. Since K is local, HK is a unique maximal ideal of K which is contained in H. Since H ∩ (K ∩ N)=H ∩ N ≤ HR and H ∩ (K ∩ N) ≤ K then HR ∩ K ≤ HK. Hence H is HK-max subring of K. Theorem 3.5 Let R be a ring, H be a subring of R and K be an ideal of R such that K ≤ H. Then H is a HR-max subring of R if and only if H/K is a (H/K)R/K -max subring of R/K. Proof: Let H be a HR-max subring of R. Then there exists an ideal N of R such that R = H + N and H ∩ N ≤ HR. By Lemma 2.4 , R/K =(H/K)+ ((N + K)/K). Now (H ∩ (N + K))/K =(K +(H ∩ N))/K ≤ (K + HR)/K. Since HR is a maximal ideal of R contained in H,(K + HR)/K is a maximal ideal of R/K contained in H/K. Then (H/K)R/K =(K + HR)/K. Therefore (H/K) ∩ ((N + K)/K) ≤ (H/K)R/K .SoH/K is a (H/K)R/K - max subring of R/K. Conversely let H/K be a (H/K)R/K - max subring of R/K. Then there exists an ideal N/K of R/K such that R/K =(H/K)+(N/K) and (H/K) ∩ (N/K) ≤ (H/K)R/K . By Lemma 2.4, R = H + N. It is clear that H ∩ N ≤ HR therefore H is a HR-max subring of R. Corollary 3.6 Let R be a ring and H be an arbitrary proper ideal of R. Then H is a HR-max subring of R if and only if the factor ring R/H is simple. Corollary 3.7 Let R be a ring with identity. Then every proper ideal of R is contained in a HR-max subring of R. Corollary 3.8 Let R be a ring. If an element e of Z(R) is idempotent, then Re is a HR-max subring of R. Corollary 3.9 Let R be a regular ring. Then every finitely generated ideal I of R is IR-max subring of R. 1236 Y. Aydın and A. Pancar 4 Conclusion We attain some properties in an associated ring R with such subrings. We show that every maximal ideal of a ring R is also HR-max subring. Also we show that if H is a HR-max subring satisfying H ≤ K ≤ R, where K is a local subring of R, then H is a HK-max subring of K and finally we prove that the class of HR-max subrings of a ring R is closed under formation of images. ACKNOWLEDGEMENTS. We would like to thank Dr. Burcu NIS˙¸ANCI TURKMEN¨ from university of Ondokuz Mayıs for her comments that greatly improved the paper. References [1] D. W. Sharpe, P. Vamos Injective Modules, Cambridge University Press, Cambridge, 1972. [2] F. Kasch Modules and Rings, Academic Press Inc., London, 1982. [3] J. S. Rose, A Course On Group Theory, Cambridge University Press, Cam- bridge, 1978. [4] M. Tashtoush, c-Maximal ideal of Finite Rings, International Journal of Algebra, 5 (2011), 135-138. [5] M. Tashtoush, Weakly c-Normal and cs-Normal Subgroups of Finite Groups, Jordan Journal of Mathematics and Statics (JJMS), 1 (2008), 123-132. [6] R. Wisbauer, Foundations of Modules and Rings, Gordon and Breach, Philadelphia, 1991. [7] T. W. Hungerford, Algebra, Springer- Verlag Press, New York, 1973. [8] Y. Wang, c-Normality of Groups and its properties , Journal of Algebra, 180 (1996), 954-965. Received: July, 2012.
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