Generalizations of -primary gamma-ideal of gamma-rings

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1 Global ournal of Pure and Applied athematics ISSN Volume 1, Number 1 (016), pp Research India Publications Generalizations of -primary gamma-ideal of gamma-rings ohamed Youssfi Elkettani 1 and Abdulbakee Kasem 1- Department of athematics Faculty of Sciences, University Ibn Tofail, B P 4, Kenitra, orocco Abstract let be a -ring, : ( ) { be a function where ( ) denotes the set of all -ideal of and be a mapping from ( ) into ( ) such that for all I (, I ( and for all I, (, I implies ( ( In this paper we introduced the notion of a - -primary -ideal which is a generalizations of -primary -ideals of -ring and study some of it properties 010 athematics Subject Classification: 13A15 Keywords: primary ideal, -primary ideal, -primary ideal, - -primary ideal, strongly - -primary ideal Introduction The concept of -ring has a special place among generalization of rings For example in Barnes [], Kyuno [5] and Luch [6] studied the stricture of -ring and obtained various generalization analogous to corresponding parts in rings theory Zhao [8] investigated the possibilities of a unified approach to studding such tow ideals, and introduced the notion of -primary ideals for a mapping that assigns to each ideal I an ideal ( of the same ring, such that the following conditions are satisfied: I ( and I implies ( ( In [3] un and all introduced the notion of -ideal expansions in -rings, let be a -ring with ( ) its set of -ideal A -ideal expansion is a function : (, which satisfies the following

2 436 ohamed Youssfi Elkettani and Abdulbakee Kasem condition: I ( for each -ideal I of and I implies ( (, for all -ideals I, of In [1] Anderson and Batanieh give a generalization of prime ideals, let : ( { be a function A proper ideal I of is siad to be -prime if for a, b with ab I \, either a I or b I In [9] Darani give a generalization of primary ideals, let : ( { be a function where ( ) denotes the set of all ideals of A proper ideal I of is called -primary if whenever a, b, ab I \ implies that either a I or b I So if we take ( = (resp, ( I 0 ) = 0), a -primary ideal is primary (resp, weakly primary) otivated by these generalization in [1], [9], Zhao s idea in [8] and [4] In this paper we give some more generalization of -primary -ideals of -rings and study the properties of these classes of -ideals Preliminaries In this section, we will give some basic concepts about -ring which you need later Definition 1 ([3]) Let and be two abelian groups and for all x, y and all, the conditions: 1 x y ; ( x y) z = xz yz, x( ) z = xz xz, x y z) = xy xz ; 3 ( x y) z = x yz) ; are satisfied, then we call a -ring Definition ([3]) A right (resp left) -ideal of a -ring we mean an additive subgroup U of such that U U (resp U U) If U isboth a right and a left -ideal, then we say that U is a -ideal of Definition 3 ([]) A proper -ideal I of is prime if a a I or b I for all a, b a b I implies Definition 4 ([3]) Let be a -ring A mapping : of -rings is called a -ring homomorphism if it satisfies: 1 a = for all a, b ; ( a = for all a, b and Definition 5 ([3]) A proper -ideal I of is primary if a a b I implies n1 a I or b I for all a, b where I := { x ( x ) x I for some n N n1 and, and ( x ) x = x where n = 1

3 Generalizations of -primary gamma-ideal of gamma-rings 437 Definition 6 ([3]) Let be a -ring with ( ) its set of -ideal A -ideal expansion is a function : (, which satisfies the following condition: (1) I ( for each -ideal I of () I implies ( ( for all -ideals I, of Example 1 ([3, Example 3]) 1 The identity function 1 : ( ) ) is a -ideal expansion of d Denote := { I and is a maximal -ideal of } A function g : ( given by g( = ( for all I ) is a -ideal expansion of 3 The constant function c : ( such that c ( =, is -ideal expansion of Definition 7 Given a -ideal expansion of A proper -ideal I of is -primary if a b I implies a I or b ( for all a, b Example ([3, Example 5]) Every -ideal I ) is c -primary, where c is a -ideal expansion of in example 1(3) ain Results In this section we extend the concept of -primary -ideal of -ring and we shall show the extend -primary -ideal enjoy analogy many of the properties -primary -ideal of -ring Definition 3 1 Let be a -ring and let : ( { be a function such that ( I, and be an expansion -ideal of A proper -ideal I of is called - -primary provided that for a, b, a b I \ implies a I or b ( Example 3 Let be a -ring Define the map : ( { as follows: 1 : defines -primary -ideals = 0 : = 0 defines weakly -primary -ideals 3 : ( = II defines almost -primary -ideals n1 4 ( n ) : = ( II ) I defines n-almost -primary -ideals n n1 5 : = ( II ) I defines - -primary -ideals n=1 6 : ( I ) = I defines any -ideals 1

4 438 ohamed Youssfi Elkettani and Abdulbakee Kasem We will begin by giving preliminary theorem from which will show some of the relations between the definitions given in Example 3 Note that the following Theorem is an extension of [9, Lemma 5] and [9, Theorem 6] The proof of Theorem 3 3, we need the following Lemma Recall that An expansion -ideal is said to be global if for any -ring homomorphism 1 1 f : R S, we have ( f ( ) = f ( ( ) for all I ) Lemma 3 Let be an expansion -ideal, and is global then ( I / = ( / I, are -ideals of If Proof Let i : / be the natural quotient homomorphism, since is global ( I/ = ( i ( ) = i( ( ) = ( / as desired Theorem 3 3 let be a -ring, and let : ( { be a function and be a global expansion -ideal The following assertions hold (i) If is global Then an -ideal I of is - -primary if and only if I/ ( is a weakly -primary of / ( (ii) If 1, : ( { be function with 1 Then if I is 1 - -primary, it is - -primary too (iii) I -primary I weakly -primary I - -primary I (n+1) almost -primary I n-almost -primary I almost -primary (iv) I is - -primary if and only if I is n-almost -primary for all n Proof (i) Assume that I is -primary -ideal of let a, b such that 0 ( a ) b ) I/ then a bi \ implies a I or b (, Hance a I/ or b ( /, So a I/ or b ( I/ ) by Lemma 3 consequently I/ ( is a weakly -primary of / ( conversely, assume that I/ ( is a weakly -primary of / ( let a, b, ab I \ then 0 ( a ) b ) I/ since I/ ( is a weakly -primary of / ( So a I/ or b ( I/ ) = ( / by Lemma 3 Hence a I or b ( as desired (ii) Let a, b such that ab I \ ( implies a, b I \ 1(, since I is 1 - -primary -ideal of then a I or b (, as required (iii) It follows from (ii) and the fact 0 n 1 n 1 (iv) It is similar of (iii)

5 Generalizations of -primary gamma-ideal of gamma-rings 439 Let be an -ideal of and : ( { a function Define : ( / / { by ( I/ = ( / for every -ideal I ) with I (and ( I/ = if ( = ) In the following we show that if I is a - -primary -ideal of, then I/ is a - -primary -ideal of / Theorem 3 4 Let I be a proper -ideal of the -ring, and let : ( { be a function, be a global expansion -ideal Assume that I is a - -primary -ideal of Then 1 If is an -ideal of with I then I/ is a - -primary -ideal of / If in addition, and I/ is - -primary, then I is - -primary 3 If ( and I is a - -primary -ideal of, then I/ is a weakly -primary -ideal of / 4 If (, is a - -primary -ideal of and I/ is a weakly -primary -ideal of /, then I is a - -primary -ideal of Proof 1 Suppose that a, b such that ( a b I/ \ ( I/ = I/ \ ( / then a b I \ and I is - -primary, gives a I or b ( Therfore, a / I/ or b/ ( / = ( I/ By lemma 3 This shows that I/ is - -primary Assume that a b I \ for some a, b Then ( a b I/ \ ( I/ = I/ \ ( I/ Since I/ is assumed to be - -primary, we get a I/ or b ( I/ = ( / by lemma 3 consequently, either a I or b (, that is I is - -primary 3 Is a direct consequence of part(1) 4 Let ab I \ where a, b Note that a b because If a b, then either a I or a ( (, since is a - -primary -ideal If a b, then ( a b ( I/ \{0} and so either a I/ or b I ( I/ = ( / Therefore, either a I or b ( consequently I - -primary -ideal of Corollary 3 5 Let be a -ring and let : ( { be a function An -ideal of is - -primary if and only if I/ ( is a weakly -primary -ideal of / ( Proof In partes () and (3) of Theorem 3 4 set = (

6 440 ohamed Youssfi Elkettani and Abdulbakee Kasem Proposition 3 6 Let I be a -ideal of a -ring such that ( be a -primary -ideal of If I is a - -primary -ideal of, then I is an -primary -ideal of Proof Assume that a b I for some elements a, b such that a I If a b, then -primary and a I implies that b ( ) ( and so we are done When a b clearly the result follows Recall that, for two ideals I and of a -ring, the residual division of I and is defined to be the ideal I : ={ x xy I for all either y } Theorem 3 7 Let I be a proper -ideal of, let : ( { be a function and is a global expension Then the following statement are equivalent: (i) I is - -primary (ii) For every a (, ( I : = I ( : (iii) For every a (, ( I : = I or ( I : = ( ( : (iv) For the ideals A and B of, A B I \ imply A I or B ( Proof (i) ii) Assume that I is - -primary We show that I ( : ( I :, let x I imply x a I so x ( I : Let x ( : imply x a ( I so x ( I : Hence I ( : ( I : one the other hand, for every r ( I :, if r a, then r ( : otherwise from r a I \ and a ( we get r I Hence ( I : I : Then ( I : = I ( : (ii) iii) Is clear because ( I : is an -ideal of (iii) iv) Let A and B are -ideals of with A B I suppose that A I and B ( We will show that A B Let b B we have tow cases b ( or b ( case one: If b ( we have ( I : = I or ( I : = ( ( : by (iii) Now from A b AB I we have A ( I : choose a A\ I, then from a ( I : \ I and (iii)we get ( I : = ( ( : Therefore, A ( I : = ( :, that is A b So A B case tow : b (, b B ( choose b B \ ( Then b b B \ (, and hence we have A b, and A ( b b ) Let a A There a b = a b b ) ab Hence, A b So A B contradiction with assumption A B Hence A I or B (

7 Generalizations of -primary gamma-ideal of gamma-rings 441 (iv) (i) Let a b I \, where a, b Then ( ( I \ By (iv) ( I or ( ( so a I or b ( Let be a -ring, and let : ( { be a function Recall that, every -primary -ideal of is - -primary Theorem 3 3 and 3 7 provide some condition under which a - -primary -ideal is -primary Theorem 3 8 Let be a -ring, and let : ( { be a function, and let I be a - -primary of (i) If I I, then I is -primary (ii) If I is not -primary and ( I = (, then ( = ( ) Proof (i) (ii) Assume that a, b such that a b I If ab, since I is - -primary, either a I or b ( Hence we may assume that ab If ai, then there exist an element a 0 I such that aa 0 Now a a = aa ab I \ ) and I is - ( 0 0 I I a0 b ( I a0 I ( I -primary that either a or ) But ) So, either a I or b ( Similarly, if bi, we can show that either a I or b ( So we may assume that ai and bi Since II, then exist c, d I with cd Now ( a c) b d) = ab ad cb cd I \, imply that either a ci or b d ( Therefore, either a I or b ( Consequently, I is -primary Since ( I, we have ( ) ( On the other hand, it follows form part(1)that II Hence ( = ( II ) ( ) So ( ) = ( Corollary 3 9 Let I be a - -primary -ideal where 3 Then I is - -primary Proof If I is -primary, then it is - -primary By theorem 3 3(iii) Assume that I is not -primary Then I I ( III by theorem 3 8(1) Hence n ( = ( II ) 1 I for all n consequently I is n -almost- -primary for every n and hence it is - -primary by theorem 3 3(iv) Theorem 3 10 Let be a -ring, let : ( { be a function and is a global expansion Suppose that {I is a family of -ideal of such that }

8 44 ohamed Youssfi Elkettani and Abdulbakee Kasem for every every,, I ) = I ), I ) and I I ) = ( I ) If for ( ( (, I is a - -primary -ideal of that is not -primary, then I = I is a - -primary -ideal of Proof Since I is a - -primary but are not -primary, then for every ( I ) = ( I )) by Theorem 3 8 on the other hand I ) for every,, and so ( I )) ( we have I I ( I ) = ( I )) Hence ( I ) = ( I ) = ( I )) for every Let a b I \, a, b and a I ( Therefore there is a such that a I Since I is - -primary and a b I \, then b ( I ) = ( consequently I is a - -primary -ideal of Corollary 3 11 Let be -ring, and let : ( { be a function If I is - -primary -ideal of with ( = I ) and ( = ( I ), then I is - -primary Proof Let a, b such that a b I \ I ) but a n1 I If ( ab ) ab n1 all, then a b = ( ) a contradiction So ( ab ) ab I \ Since I is - -primary -ideal of and for n 1 a I, so ( b b ( Hence b ( = ( I ) So I is - -primary Next we give a definition of additive expansion ideal function An expansion ideal is called additive if ( I = ( ( for every -ideals I and of -ring Note that in Theorem 3 1 is an expansion ideal additive Theorem 3 1 Let I, are - -primary -ideals of -ring that is not -primary -ideals such that ( I = (, suppose that the two -ideals ( and ( are not coprime Then 1 ( I = ( ) If ( and ( I then I is a - -primary -ideal of Proof 1 By Theorem 3 8 we have ( = ( ) and ( = ( ) Also we have I and implies ) I = ( = ) ) = ( ) Hence ( I = ( )

9 Generalizations of -primary gamma-ideal of gamma-rings 443 Assume that ( and ( I Since (, then I is proper -ideal of, by part(1) Since ( I / I/ I and I is - -primary, we get that ( I / is a weakly -primary -ideal of / By Theorem 3 4(3) On the other hand is also - -primary, by Theorem 3 3(i) Now, the assertion follows from theorem 3 4(4) In the end of this short paper we give the following result related of strongly - -primary We called a proper strongly -ideal I of to be a - -primary -ideal of if I1 I I \ for -ideal I 1, I of implies that either I 1 I or I ( ) I Theorem 3 13 Let I be a proper -ideal of -ring Then the following conditions are equivalent: 1 I is strongly - -primary For every -ideals I 1, I of such that I I1, I1I I \ implies that either I 1 = I or I ( Proof 1 (1) () Is obviously () (1) Let, I be -ideals of such that I I \, then we have that ( I = I II I \, set I1 = I Then, by the hypothesis either I 1 I or I ( Therefore, either I or I ( So I is strongly - -primary -ideal of Corollary 3 14 Let I, are two proper -ideals of -ring Such that I is strongly - -primary and I Then is strongly - -primary Proof By take I 1 = I = in Theorem 3 13 we have I as required Acknowledgment We would like to thank the referee for a careful reading of our article and insightful comments which saved us from several errors References [1] Anderson, D D, Batanieh,, Generalizations of prime ideals Comm Algebra 36 (008),

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