A note on maximal ideals in ordered semigroups
In commutative rings having an identity element, every maximal ideal is a prime ideal, but the converse statement does not hold, in general. According to the present note, similar results for ordered semigroups and semigroups -without order- also hold. In fact, we prove that in commutative order...
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irk-123456789-1546732019-06-16T01:32:45Z A note on maximal ideals in ordered semigroups Kehayopulu, N. Ponizovskii, J. Tsingelis, M. In commutative rings having an identity element, every maximal ideal is a prime ideal, but the converse statement does not hold, in general. According to the present note, similar results for ordered semigroups and semigroups -without order- also hold. In fact, we prove that in commutative ordered semigroups with identity each maximal ideal is a prime ideal, the converse statement does not hold, in general. 2003 Article A note on maximal ideals in ordered semigroups / N. Kehayopulu, J. Ponizovskii, M. Tsingelis // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 32–35. — Бібліогр.: 3 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 06F05. http://dspace.nbuv.gov.ua/handle/123456789/154673 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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English |
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In commutative rings having an identity element,
every maximal ideal is a prime ideal, but the converse statement
does not hold, in general. According to the present note, similar
results for ordered semigroups and semigroups -without order- also
hold. In fact, we prove that in commutative ordered semigroups
with identity each maximal ideal is a prime ideal, the converse
statement does not hold, in general. |
| format |
Article |
| author |
Kehayopulu, N. Ponizovskii, J. Tsingelis, M. |
| spellingShingle |
Kehayopulu, N. Ponizovskii, J. Tsingelis, M. A note on maximal ideals in ordered semigroups Algebra and Discrete Mathematics |
| author_facet |
Kehayopulu, N. Ponizovskii, J. Tsingelis, M. |
| author_sort |
Kehayopulu, N. |
| title |
A note on maximal ideals in ordered semigroups |
| title_short |
A note on maximal ideals in ordered semigroups |
| title_full |
A note on maximal ideals in ordered semigroups |
| title_fullStr |
A note on maximal ideals in ordered semigroups |
| title_full_unstemmed |
A note on maximal ideals in ordered semigroups |
| title_sort |
note on maximal ideals in ordered semigroups |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2003 |
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http://dspace.nbuv.gov.ua/handle/123456789/154673 |
| citation_txt |
A note on maximal ideals in ordered semigroups / N. Kehayopulu, J. Ponizovskii, M. Tsingelis // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 32–35. — Бібліогр.: 3 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT kehayopulun anoteonmaximalidealsinorderedsemigroups AT ponizovskiij anoteonmaximalidealsinorderedsemigroups AT tsingelism anoteonmaximalidealsinorderedsemigroups AT kehayopulun noteonmaximalidealsinorderedsemigroups AT ponizovskiij noteonmaximalidealsinorderedsemigroups AT tsingelism noteonmaximalidealsinorderedsemigroups |
| first_indexed |
2025-07-14T06:42:10Z |
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2025-07-14T06:42:10Z |
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1837603572384006144 |
| fulltext |
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2003). pp. 32–35
c© Journal “Algebra and Discrete Mathematics”
A note on maximal ideals in ordered semigroups
N. Kehayopulu, J. Ponizovskii, M. Tsingelis
Abstract. In commutative rings having an identity element,
every maximal ideal is a prime ideal, but the converse statement
does not hold, in general. According to the present note, similar
results for ordered semigroups and semigroups -without order- also
hold. In fact, we prove that in commutative ordered semigroups
with identity each maximal ideal is a prime ideal, the converse
statement does not hold, in general.
There is an important class of ideals of rings which are prime, namely,
the maximal ideals. In fact, in a commutative ring with identity every
maximal ideal is a prime ideal. On the other hand, there are rings poss-
esing a nontrivial prime ideal which is not maximal (cf. e.g. [1]). Similar
results for ordered semigroups, also for semigroups -without order- also
hold.
If (S, .,≤) is an ordered semigroup, a non-empty subset I of S is called
a left (resp. right) ideal of S if 1) SI ⊆ I (resp. IS ⊆ I) and 2) a ∈ I,
S ∋ b ≤ a implies b ∈ I [2]. If (S, .) is a semigroup, a left (resp. right)
ideal of S is a non-empty subset I of S such that SI ⊆ I (resp. IS ⊆ I).
If S is a semigroup or an ordered semigroup and I both a left and a
right ideal of S, then it is called an ideal of S. An ideal I of a semigroup
(resp. ordered semigroup) S is called prime if a, b ∈ S such that ab ∈ I
implies a ∈ I or b ∈ I. Equivalent Definition: A, B ⊆ S such that
AB ⊆ I implies A ⊆ I or B ⊆ I [2]. An ideal M of a semigroup or an
ordered semigroup S is called proper if M 6= S [3]. A proper ideal M of
a semigroup or an ordered semigroup S is called maximal if there exists
no ideal T of S such that M ⊂ T ⊂ S, equivalently, if for each ideal T
of S such that M ⊆ T , we have T = M or T = S (cf. also [2]). If S is
an ordered semigroup and H ⊆ S, we denote
(H] := {t ∈ S | t ≤ h for some h ∈ H}.
2000 Mathematics Subject Classification: 06F05.
Key words and phrases: maximal ideal, prime ideal in ordered semigroups.
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h.N. Kehayopulu, J. Ponizovskii, M. Tsingelis 33
If S is an ordered semigroup (resp. semigroup) and ∅ 6= A ⊆ S, we
denote by I(A) the ideal of S generated by A i.e. the smallest -under
inclusion relation- ideal of S containing A. For an ordered semigroup S,
we have I(A) = (A ∪ SA ∪ AS ∪ SAS] (cf. [2]). For a semigroup S, we
have I(A) = A ∪ SA ∪ AS ∪ SAS.
Let {(Si, ◦i,≤i) | i ∈ I} be a non-empty family of ordered semigroups.
The cartesian product
∏
i∈I
Si with the multiplication “∗” and the order
“¹” on
∏
i∈I
Si defined by
∗ :
∏
i∈I
Si ×
∏
i∈I
Si →
∏
i∈I
Si | ((xi)i∈I , (yi)i∈I) → (xi)i∈I ∗ (yi)i∈I where
(xi)i∈I ∗ (yi)i∈I := (xi ◦i yi)i∈I
¹: =
{
((xi)i∈I , (yi)i∈I) ∈
∏
i∈I
Si ×
∏
i∈I
Si | xi ≤i yi ∀ i ∈ I
}
is an ordered semigroup.
In the following we consider the
∏
i∈I
Si as the ordered semigroup with the
multiplication and the order defined above.
Lemma 1. Let {(Si, ◦i,≤i) | i ∈ I} be a family of ordered semigroups.
If Ji is an ideal of Si for every i ∈ I, then the set
∏
i∈I
Ji is an ideal of
∏
i∈I
Si.
Proof. 1) ∅ 6=
∏
i∈I
Ji ⊆
∏
i∈I
Si (since Ji 6= ∅ ∀ i ∈ I).
2)
∏
i∈I
Si ∗
∏
i∈I
Ji ⊆
∏
i∈I
Ji. In fact:
Let (xi)i∈I ∈
∏
i∈I
Si and (yi)i∈I ∈
∏
i∈I
Ji. Since xi ∈ Si and yi ∈ Ji for
every i ∈ I, we have xi ◦i yi ∈ Si ◦i Ji ⊆ Ji for every i ∈ I. Then we have
(xi)i∈I ∗ (yi)i∈I := (xi ◦i yi)i∈I ∈
∏
i∈I
Ji
.
3) Let (yi)i∈I ∈
∏
i∈I
Ji and
∏
i∈I
Si ∋ (xi)i∈I ¹ (yi)i∈I . Then (xi)i∈I ∈
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h.34 A note on maximal ideals in ordered semigroups
∏
i∈I
Ji.
Indeed: Since yi ∈ Ji, Si ∋ xi ≤i yi and Ji is an ideal of Si for every
i ∈ I, we have xi ∈ Ji for every i ∈ I. Then (xi)i∈I ∈
∏
i∈I
Ji. Similarly,
the set of
∏
i∈I
Ji is a right ideal of
∏
i∈I
Si. 2
In the following, we denote by S the closed interval [0, 1] of real numbers.
The set S := [0, 1] with the usual multiplication- order “.” and “≤” is an
ordered semigroup.
Lemma 2. If a ∈ S, then the set Ia := [0, a] is an ideal of S.
Proof. First of all ∅ 6= Ia ⊆ S (since a ∈ [0, a]). Let x ∈ S, y ∈ Ia. Since
0 ≤ x ≤ 1, 0 ≤ y ≤ a, we have 0 ≤ xy ≤ 1a = a. Then xy ∈ Ia. Let
y ∈ Ia and S ∋ x ≤ y. Since 0 ≤ x, y ≤ a and x ≤ y, we have 0 ≤ x ≤ a.
Then x ∈ Ia. Similarly, the set Ia is a right ideal of S. 2
Theorem. Let (S, .,≤) be a commutative ordered semigroup with iden-
tity. If M is a maximal ideal of S, then M is a prime ideal of S. The
converse statement does not hold, in general.
Proof. Let e be the identity of S, and M a maximal ideal of S.
Let a, b ∈ S, ab ∈ M , a /∈ M . Then b ∈ M . In fact:
Since S is commutative, we have
I(M ∪ {a}) = ((M ∪ {a}) ∪ S(M ∪ {a}) ∪ (M ∪ {a})S ∪ S(M ∪ {a})S]
= ((M ∪ {a}) ∪ S(M ∪ {a}) ∪ S2(M ∪ {a})].
Since M ∪ {a} = e(M ∪ {a}) ⊆ S(M ∪ {a}), we have
S(M ∪ {a}) ⊆ S2(M ∪ {a}) ⊆ S(M ∪ {a}),
then S(M ∪ {a}) = S2(M ∪ {a}).
Hence we have
I(M ∪ {a}) = (S(M ∪ {a})]..........(∗)
On the other hand, M ⊂ M ∪ {a} ⊆ I(M ∪ {a}) (since a /∈ M). Since
I(M∪{a}) is an ideal and M a maximal ideal of S, we have I(M∪{a}) =
S, and e ∈ (S(M ∪{a})] by (*). Then there exist x ∈ S and y ∈ M ∪{a}
such that e ≤ xy. Then b = eb ≤ xyb. If y ∈ M , then xyb ∈ SMS ⊆ M ,
and b ∈ M . If y = a, then b ≤ x(ab) ∈ SM ⊆ M , and b ∈ M .
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h.N. Kehayopulu, J. Ponizovskii, M. Tsingelis 35
For the converse statement, we consider the ordered semigroup S := [0, 1]
and the ordered semigroup (S × S, ∗ ¹) constructed above. The set
(S × S, ∗,¹) is a commutative ordered semigroup and the element (1, 1)
is the identity element of S × S. Let
T := S × {0}(= [0, 1] × {0}).
Clearly S is an ideal of S. By Lemma 2, the set I0(= {0}) is an ideal of
S. Then, by Lemma 1, the set T := S × {0} is an ideal of S × S.
The set T is a prime ideal of S × S. In fact:
Let (x, y), (z, w) ∈ S × S, (x, y) ∗ (z, w) ∈ T . Since (x, y) ∗ (z, w) :=
(xz, yw) ∈ T := S × {0}, we have yw = 0, then y = 0 or w = 0. Then
(x, y) ∈ S × {0} := T or (z, w) ∈ S × {0} := T .
The set T is not a maximal ideal of S ×S. Indeed: By Lemma 2, the set
[0, 1/2] := I1/2 is an ideal of S. By Lemma 1, the set S × [0, 1/2] is an
ideal of S×S. On the other hand, T := S×{0} ⊂ S×[0, 1/2] ⊂ S×S. 2
This is part of our research work supported by the Ministry of Devel-
opment, General Secretariat of Research and Technology -International
Cooperation Division (Greece-Russia), Grant No. 70/3/4967.
References
[1] D. M. Burton, A First Course in Rings and Ideals, Addison-Wesley, 1970.
[2] N. Kehayopulu, On weakly prime ideals of ordered semigroups, Mathematica
Japonica, 35 (No. 6) (1990), 1051-1056.
[3] N.Kehayopulu, Note on Green’s relations in ordered semigroups, Mathematica
Japonica, 36 (No. 2) (1991), 211-214.
Contact information
N. Kehayopulu,
M. Tsingelis
University of Athens, Department of Math-
ematics Section of algebra and geome-
try, Panepistemiopolis, Athens 157 84,
GREECE
E-Mail: nkehayp@cc.uoa.gr
J. Ponizovskii Russian State Hydrometeorological Uni-
versity Department of Mathematics Mal-
ookhtinsky pr. 98 195196, Saint-Petersburg,
Russia
Received by the editors: 06.12.2002.
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