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