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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

On f - prime radical in ordered semigroups

Ze Gu
Published Online: 2018-06-07 | DOI: https://doi.org/10.1515/math-2018-0053

Abstract

In this paper, we introduce the concepts of f-prime ideals, f-semiprime ideals and f-prime radicals in ordered semigroups. Furthermore, some results on f-prime radicals and f-primary decomposition of an ideal in an ordered semigroup are obtained.

Keywords: Ordered semigroup; f-prime ideal; f-semiprime ideal; f-prime radical; f-primary decomposition

MSC 2010: 06F05; 20M10

1 Introduction and preliminaries

Prime radical theorem is an important result in commutative ring theory and commutative semigroup theory (see [1,2]). In [3], Hoo and Shum gave a similar result in a residuated negatively ordered semigroup. Wu and Xie extended the result to a commutative ordered semigroup by using m-systems in [4]. In [5], Tang and Xie characterized in detail the radicals of ideals in ordered semigroups. In 1969, Murata, Kurata and Marubayashi introduced the notions of f-prime ideals and f-prime radicals in ring theory (see [6]), which generalized the concepts of prime ideals and prime radicals. In [7], Sardar and Goswami extended these concepts and results of ring theory to semirings. In this paper, we introduce the concepts of f-prime ideals and f-prime radicals in ordered semigroups and extend some results of rings and semirings to ordered semigroups. We also introduce the notion of f-semiprime ideals in ordered semigroups and obtain the result that the f-prime radical of an ideal I in an ordered semigroup is the least f-semiprime ideal containing I.

Next we list some basic concepts and notations on ordered semigroups (see [8]). An ordered semigroup is a semigroup (S, ·) endowed with an order relation “ ≤ ” such that

(a,b,xS)abxaxbandaxbx.

Let (S, ·, ≤) be an ordered semigroup. A non-empty subset I of S is called an ideal of S if it satisfies the following conditions: (1) SIISI; (2) aI and bS, ba implies bI. An ideal I of S is called weakly prime if ABI implies AI or BI for any ideals A, B of S; an ideal I of S is called weakly semiprime if A2I implies AI for any ideal A of S. For hS, we denote

(h]={tSth};[h)={sShs}.

A subset M of S is called an m-system of S, if for any a, bM, there exists xS such that (axb] ∩ M ≢ ∅. A subset N of S is said to be a n-system of S if for any aN, there exists xS such that [axa) ⊆ N.

2 f-prime ideals and f-prime radical of an ideal

Definition 2.1

Let S be an ordered semigroup. Denote by I(S) the set of all ideals of S. Define a mapping f: SI(S) which satisfies the following conditions

  1. af(a);

  2. xf(a) ∪ I implies that f(x) ⊆ f(a) ∪ I for any II(S).

We call such mapping f a good mapping on S.

Example 2.2

Let S be an ordered semigroup. If f(a) = I(a) for all aS, where I(a) is the principal ideal generated by a, then it is easy to see that f satisfies the above conditions.

Example 2.3

We consider the ordered semigroup S = {a, b, c} defined by multiplication and the order below:

abcaaaabaabcabb

≤:={(a,a),(a,b),(a,c),(b,b),(b,c),(c,c)}.

The ideals of S are the sets:

{a},{a,b}andS.

If we define f(a) = {a}, f(b) = {a, b} and f(c) = S, then it is easy to see that f satisfies the above conditions.

Definition 2.4

Let S be an ordered semigroup and f a good mapping on S. A subset F of S is called an f-system of S if F contains an m-system F, called the kernel of F, such that f(t) ∩ F ≢ ∅ for any tF.

Especially, ∅ is also defined to be an f-system.

Remark 2.5

  1. Every m-system is an f-system with kernel itself.

  2. If F is an f-system with kernel F, then F = ∅ if and only if F = ∅.

Definition 2.6

Let S be an ordered semigroup and f a good mapping on S. An ideal I of S is called f-prime if its complement C(I) in S is an f-system.

Remark 2.7

As we know, the complement of a weakly prime ideal of an ordered semigroup is an m-system. Furthermore, every m-system is an f-system. Hence, every weakly prime ideal of an ordered semigroup is an f-prime ideal. But the converse is not true in general.

Example 2.8

We consider the ordered semigroup S in Example 2.3. Let I = {a}. Then C(I) = {b, c} is an f-system with kernel F = {b}. Hence, I is an f-prime ideal. However, I is not weakly prime. Indeed: {a, b} is an ideal of S and {a, b}{a, b} ⊆ I, but {a, b} is not contained in I.

Proposition 2.9

Let P be an f-prime ideal of an ordered semigroup S and a, bS. If f(a)f(b) ⊆ P, then either aP or bP.

Proof

Suppose that a, bC(P). Since P is an f-prime ideal, C(P) is an f-system. Hence, f(a) ∩ [C(P)] ≠ ∅ and f(b) ∩ [C(P)] ≠ ∅, where [C(P)] is the kernel of C(P). Let x1f(a) ∩ [C(P)] and x2f(b) ∩ [C(P)]. Since [C(P)] is an m-system, (x1rx2] ∩ [C(P)] ≠ ∅ for some rS. Thus (x1rx2] ∩ C(P) ≠ ∅. Also x1rx2f(a)f(b) ⊆ P. Thus (x1rx2] ⊆ P which is a contradiction. Therefore, either aP or bP. □

Corollary 2.10

Let P be an f-prime ideal of an ordered semigroup S and aiS (i = 1, 2, ⋯, n). If f(a1)f(a2) ⋯ f(an) ⊆ P, then aiP for some i.

Let S be an ordered semigroup. Denote by fPI(S) and fS(S) the set of all f-prime ideals of S and the set of all f-systems of S respectively.

Definition 2.11

Let S be an ordered semigroup and I be an ideal of S. We call the set {aS|(∀ FfS(S)) aFFI ≠ ∅} the f-prime radical of I, denoted by rf(I).

Theorem 2.12

Let S be an ordered semigroup and I be an ideal of S. Then rf(I) = ∩PΓP where Γ = {PfPI(S)|IP}.

Proof

Let J = ∩PΓP where Γ = {PfPI(S)|IP}. If xJ, then there exists an f-prime ideal P containing I such that xP. Thus C(P) is an f-system containing x but C(P) ∩ I = ∅. Hence xrf(I). Therefore, rf(I) ⊆ J. On the other hand, if yrf(I), then there exists an f-system F containing y such that FI = ∅. Thus yC(F) and C(F) is an f-prime ideal such that IC(F). Hence yJ and so Jrf(I). Therefore, rf(I) = J. □

Corollary 2.13

Let S be an ordered semigroup and I be an ideal of S. Then rf(I) is an ideal of S.

3 f-semiprime ideals

In this section, we introduce the concept of f-semiprime ideals in ordered semigroups and obtain that the f-prime radical of an ideal I is the least f-semiprime ideal containing I.

Definition 3.1

Let S be an ordered semigroup. A subset A of S is said to be an fn-system of S if A=iΓFi where {Fi|iΓ} ⊆ fS(S).

Definition 3.2

An ideal P of an ordered semigroup S is said to be f-semiprime if its complement C(P) in S is an fn-system.

Clearly, every f-system is an fn-system. Therefore, every f-prime ideal is an f-semiprime ideal.

Proposition 3.3

Let S be an ordered semigroup and I a weakly semiprime ideal of S. Then I is an f-semiprime ideal of S.

Proof

Since I is a weakly semiprime ideal, C(I) is a n-system. Thus C(I) is the union of some m-systems of S. Since every m-system is an f-system, C(I) is the union of some f-systems of S. Hence C(I) is an fn-system. Therefore, I is an f-semiprime ideal. □

Proposition 3.4

Let P be an f-semiprime ideal of an ordered semigroup S and aS. If f(a)f(a) ⊆ P, then aP.

Proof

Suppose that aC(P). Since P is an f-semiprime ideal, C(P) is an fn-system. Thus C(P)=iΓFi, where {Fi| iΓ} ⊆ fS(S). Hence, aFi for some iΓ. Therefore, f(a) ∩ Fi ≠ ∅, where Fi is the kernel of Fi. Let xf(a) ∩ Fi . Since Fi is an m-system, (xrx] ∩ Fi ≠ ∅ for some rS. Thus (xrx] ∩ Fi ≠ ∅ and so (xrx] ∩ C(P) ≠ ∅. Also xrxf(a)f(a) ⊆ P. Therefore, (xrx] ⊆ P which is a contradiction. Hence aP. □

Proposition 3.5

Let S be an ordered semigroup and I be an ideal of S. Then rf(I) is an f-semiprime ideal of S.

Proof

By Theorem 2.12, we know that rf(I) = ∩PΓP where Γ = {PfPI(S)| IP}. Thus C(rf(I)) = ⋃PΓ C(P). Since every C(P) is an f-system, C(rf(I)) is an fn-system. Therefore, rf(I) is an f-semiprime ideal of S. □

Proposition 3.6

Let S be an ordered semigroup and I be an f-semiprime ideal of S. Then rf(I) = I.

Proof

By Theorem 2.12, we have Irf(I). Let xI. Since C(I) is an fn-system, C(I)=iΓFi where {Fi| iΓ} ⊆ fS(S). Thus xiΓFi and so xFi for some iΓ, but FiI = ∅. Hence Fi is an f-system containing x but not containing any element of I. By Definition 2.11, xrf(I). It follows that rf(I) ⊆ I. Therefore rf(I) = I. □

Definition 3.7

Let S be an ordered semigroup and f a good mapping on S. Let aS and II(S). The set {xS | f(a)f(x) ⊆ I}, denoted by I : a, is called the left f-quotient of I by a. Moreover, for any ideal J of S, the left f-quotient of I by J is defined to be aJ(I:a), denoted by I : J.

Remark 3.8

Similar to Definition 3.7, we call the set {xS| f(x)f(a) ⊆ I} right f-quotient of I by a, denoted by a : I. Moreover, aJ(a:I) is defined to be right f-quotient of I by J. In what follows, unless otherwise mentioned, f-quotient means left f-quotient.

We note that I : a may be empty. See the following example.

Example 3.9

We consider the ordered semigroup S of Example 2.3. If we define f(a) = f(b) = f(c) = S, then the mapping f is a good mapping. Let I = {a}. Then I : a, I : b and I : c are all empty.

The following result can be easily obtained from the above definition.

Proposition 3.10

Let S be an ordered semigroup and f a good mapping on S. If I, I, I, J, J, JI(S) and aS, then

  1. III : aI : a and I : JI : J;

  2. JJI : JI : J;

  3. (II) : a = (I : a) ∩ (I : a) and (II) : J = (I : J) ∩ (I : J).

Proposition 3.11

Let S be an ordered semigroup and f a good mapping on S. If II(S) and aS, then I : a is either empty or an ideal containing I of S.

Proof

Suppose that I : a ≠ ∅. Let xI : a and rS. Then rx, xrf(x). Thus f(rx) ⊆ f(x) and f(xr) ⊆ f(x). Also f(a)f(x) ⊆ I. Therefore, f(a)f(rx) ⊆ I and f(a)f(xr) ⊆ I. Let zyI : a. Then zf(y) and f(a)f(y) ⊆ I. Thus f(z) ⊆ f(y). Therefore, f(a)f(z) ⊆ I, which implies that zI : a. Hence, I : a is an ideal of S. Next we prove that II : a.

Let bI and xI : a. Then bf(x) ∪ I. Thus f(b) ⊆ f(x) ∪ I. Also f(a)f(x) ⊆ I. It follows that f(a)f(b) ⊆ f(a)(f(x) ∪ I) = f(a)f(x) ∪ f(a)II. Hence bI : a and so II : a. □

Let S be an ordered semigroup and f a good mapping on S. Denote the following condition by (α):

(FfS(S))(II(S))FIFI.

If f(a) = I(a) for every aS, then S satisfies the condition (α). But this is not true for any good mapping f. See the following example.

Example 3.12

We consider the ordered semigroup S = {a, b, c} defined by multiplication and the order below:

abcaaaabaabcabc

≤:={(a,a),(a,b),(a,c),(b,b),(b,c),(c,c)}.

It is easy to check that S is an ordered semigroup. The ideals of S are the sets:

{a},{a,b}andS.

If we define f(a) = {a}, f(b) = f(c) = S, then it is easy to see that f is a good mapping. Let I = {a, b} and F = {b, c}. Then F is an f-system with kernel F* = {c} and FI ≠ ∅. However, F*I = ∅.

Proposition 3.13

Let S be an ordered semigroup and f a good mapping on S. If I, JI(S), then

  1. IJrf(I) ⊆ rf(J);

  2. rf(rf(I)) = rf(I);

  3. rf(IJ) = rf(rf(I) ∪ rf(J));

  4. rf(IJ) = rf(I) ∩ rf(J), if S satisfies the condition (α).

Proof

(1) and (2) are obvious.

(3) From Theorem 2.12, we have IJrf(I)∪rf(J). By condition (1), we have rf(IJ) ⊆ rf(rf(I)∪rf(J)) and rf(I)∪rf(J) ⊆ rf(IJ). Combining conditions (1) and (2), we obtain rf(rf(I) ∪ rf(J)) ⊆ rf(rf(IJ)) = rf(IJ).

(4) It is obvious that rf(IJ) ⊆ rf(I) ∩ rf(J) from condition (1). Next we prove the other inclusion. Let xrf(I) ∩ rf(J) and F be an f-system containing x. Suppose that aFI and bFJ. By assumption (α), there exist a*F*I and b*F*J. Since F* is an m-system, (a*zb*] ∩ F* ≠ ∅ for some zS. Thus (a*zb*] ∩ F ≠ ∅. Moreover, a*zb*IJ and so (a*zb*] ⊆ IJ. Hence F ∩ (IJ) ≠ ∅. It follows that xrf(IJ). Therefore rf(IJ) = rf(I) ∩ rf(J). □

Combining Proposition 3.5, 3.6 and 3.13 (1), we have the following result.

Theorem 3.14

Let I be an ideal of an ordered semigroup S. Then rf(I) is the least f-semiprime ideal containing I.

4 f-primary decomposition of an ideal

In this section, we introduce the concepts of f-primary ideals and f-primary decomposition of an ideal in ordered semigroups, and conclude that the number of f-primary components and the f-prime radicals of f-primary components of a normal decomposition of an ideal I depend only on I under some assumptions.

Definition 4.1

Let S be an ordered semigroup and f a good mapping on S. An ideal I of S is called left f-primary if f(a)f(b) ⊆ I implies that arf(I) or bI.

Remark 4.2

By symmetry, we call an ideal I of S f-primary if f(a)f(b) ⊆ I implies that aI or brf(I). In what follows unless otherwise mentioned, f-primary means left f-primary.

By Proposition 2.9, we note that f-prime ideals must be f-primary ideals.

From Definition 4.1, we obtain easily the following result.

Proposition 4.3

Let S be an ordered semigroup satisfying the condition (α). If I and I are f-primary ideals of S such that rf(I) = rf(I), then I = II is also an f-primary ideal of S such that rf(I) = rf(I) = rf(I).

Let S be an ordered semigroup and f a good mapping on S. Denote the following condition by (β):

(I,JI(S))Jrf(I)I:J.

Theorem 4.4

Let S be an ordered semigroup satisfying the condition (β). An ideal I of S is f-primary if and only if I : J = I for all ideals Jrf(I).

Proof

Suppose that I is an f-primary ideal of S and J is an ideal of S not contained in rf(I). Since S satisfies the condition (β), I : J ≠ ∅. Thus I : b ≠ ∅ for all bJ. Hence II : b for every bJ and so II : J. Now we choose an element cJrf(I). By the condition (β), I : c ≠ ∅. Moreover, f(c)f(a) ⊆ I for any aI : c. Since I is f-primary and crf(I), aI. Thus I : cI. Therefore, I = I : c and so I : JI : c = I. Consequently, I = I : J.

Conversely, suppose that I : J = I for all ideals J not contained in rf(I). Let f(a)f(b) ⊆ I and arf(I). Since af(a), f(a) is not a subset of rf(I) and so I : f(a) = I. For any af(a), f(a) ⊆ f(a). Thus f(a)f(b) ⊆ f(a)f(b) ⊆ I and thus bI : f(a) = I. It follows that I is f-primary. □

Definition 4.5

If an ideal I of an ordered semigroup S can be written as I = I1I2 ∩ ⋯ ∩ In where each Ii is an f-primary ideal, then this is called an f-primary decomposition of I and each Ii is called the f-primary component of the decomposition.

A f-primary decomposition in which no Ii contains the intersection of the remaining Ij is called irredundant. Moreover, an irredundant f-primary decomposition in which the radicals of the various f-primary components are all different is called a normal decomposition.

From Proposition 4.3, we note that each f-primary decomposition can be refined into one which is normal.

Let S be an ordered semigroup and f a good mapping on S. Denote the following condition by (γ): if for any f-primary ideal I of S, we have I : I = S. If f(a) = I(a) for all aS, then S satisfies the condition (γ). But the condition (γ) need not be satisfied for any good mapping f.

Example 4.6

From Example 2.8, P = {a} is an f-prime ideal of S and so P is f-primary. However P : P = {a} ≠ S.

Theorem 4.7

Let S be an ordered semigroup satisfying the conditions (α), (β) and (γ). If an ideal A of S has two normal f-primary decompositions A=i=1nIi=i=1mIi, then n = m and rf(Ii) = rf( Ii) for 1 ≤ in = m by a suitable ordering.

Proof

It is easy to see that the result holds in the case A = S. Next we prove the case that AS, where all f-primary components I1, ⋯, In, I1,,Im are proper ideals. We may assume that rf(I1) is maximal in the set {rf(I1), ⋯, rf(In), rf( I1),

⋯, rf( Im)}. Now we prove that rf(I1) = rf( Ii) for some i. It is enough to show that I1rf( Ii). Suppose that I1rf( Ii) for all 1 ≤ im. Then we have, by Theorem 4.4, Ii:I1=Ii for all 1 ≤ im, and so

A:I1=(I1I2In):I1=(I1:I1)(I2:I1)(In:I1)=I1I2In=A.

If n = 1, then, by the condition (γ), S = I1 : I1 = A : I1 = A which is a contradiction. If n > 1, then, by the condition (γ) and the fact that I1rf(Ii) for all 2 ≤ in, we have

A=A:I1=(I1I2In):I1=(I1:I1)(I2:I1)(In:I1)=I2In.

This is also a contradiction. By a suitable ordering, we may have rf(I1) = rf( I1).

We use an induction on the number n of f-primary components. If n = 1, then A=I1=i=1mIi. Suppose that m > 1. Then I1rf(Ii) for all 2 ≤ im. Since S=I1:I1=i=1m(Ii:I1), we have, by Theorem 4.4, S=I2==Im which is a contradiction. Thus m = 1 = n. Now let us suppose that the conclusions hold for the ideals which are represented by fewer than n f-primary components. Let I=I1I1. Then I is an f-primary ideal such that rf(I) = rf(I1) = rf( I1) from Proposition 4.3. By the condition (γ), we have S = I1 : I1I1 : I and thus I1 : I = S. From the fact that Irf(Ii) for all 2 ≤ in, we obtain Ii : I = Ii. Hence A:I=i=2nIi. Similarly, we can show that A:I=i=2mIi. Consequently, A:I=i=2nIi=i=2mIi and the decompositions are both normal. By the induction hypothesis, we have n – 1 = m – 1 and so n = m. Moreover, by a suitable ordering, we have rf(Ii) = rf( Ii) for all 2 ≤ in = m. □

Acknowledgement

This work is supported by the National Natural Science Foundation of China (No. 11701504), the Young Innovative Talent Project of Department of Education of Guangdong Province (No. 2016KQNCX180) and the University Natural Science Project of Anhui Province (No. KJ2018A0329).

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About the article

Received: 2018-02-06

Accepted: 2018-04-25

Published Online: 2018-06-07


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 574–580, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0053.

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