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On some powerful ideals of semigroups. (English) Zbl 1382.20054

Summary: Let \(S\) be a non trivial additive cancellation commutative semi-group having a zero 0 with the quotient group \(G=\{s_1-s_2\mid s_1,s_2\in S\}\). A prime ideal \(P\) of \(S\) is called strongly prime if for \(a,b\in G\), whenever \(a+b\in P\), we get \(a\in P\) or \(b\in P\). We show that a prime ideal of \(S\) is strongly prime \(\Leftrightarrow\) it is powerful. Finally, we prove that if \(O\) is an oversemigroup of \(S\) and \(S\) and \(O\) have the nonzero ideal \(I\) where \(I\) is powerful in \(O\), then \(3I\) is a powerful ideal of \(S\).

MSC:

20M12 Ideal theory for semigroups
20M14 Commutative semigroups
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