A source of semiprimeness on inverse and completely regular semigroups

Rasie Mekera; Didem Yeşil

doi:10.3842/umzh.v76i8.7699

Rasie Mekera Department of Mathematics, Faculty of Science, Çanakkale Onsekiz Mart University, Turkey
Didem Yeşil Department of Mathematics, Faculty of Science, Çanakkale Onsekiz Mart University, Turkey

DOI: https://doi.org/10.3842/umzh.v76i8.7699

Keywords: Regular semigroup, completely regular semigroups, inverse semigroup, semiprime semigroup

Abstract

UDC 512.5

We define $|S_{S}|$-inverse semigroup and $|S_{S}|$-completely regular semigroup structures that are not encountered in the literature and are studied in our work for the first time. The properties of these new semigroup types and their relations with the other semigroup types, such as $|S_{S}|$-idempotent, $|S_{S}|$-regular, $|S_{S}|$-nonzero, and $|S_{S}|$-reduced semigroups, are discussed. Further, the relationship between the source of semiprimeness and the semiprime ideal is analyzed. It is also shown that the intersection of all semiprime ideals is equal to the source of semiprimeness. Moreover, if a function is surjective, then it can be shown that the analyzed two structures are preserved under homomorphism.

References

R. Giri, J. Kim, M. Sohn, Semiprime ideals in semigroups, Math. Japon, 33, 269–273 (1988).

Y. S. Park, J. P. Kim, Prime and semiprime ideals in semigroups, Kyungpook Math. J., 32, № 3, 629–633 (1992).

Š. Schwarz, Prime ideals and maximal ideals in semigroups, Czechoslovak Math. J., 19, № 1, 72–79 (1969). DOI: https://doi.org/10.21136/CMJ.1969.100877

N. Aydın, C. Demir, D. K. Camcı, The source of semiprimeness of rings, Commun. Korean Math. Soc., 33, № 4, 1083–1096 (2018).

B. Albayrak, D. Yešil, D. Karalarlıoğlu Camcı, The source of semiprimeness of semigroups, J. Math., 2021, 1–8 (2021). DOI: https://doi.org/10.1155/2021/4659756

J. Ježek, T. Kepka, P. Nĕmec, Commutative semigroups that are nil of index 2 and have no irreducible elements, Math. Bohem., 133, № 1, 1–7 (2008). DOI: https://doi.org/10.21136/MB.2008.133941

A. Reinhart, Structure of general ideal semigroups of monoids and domains, J. Commut. Algebra, 4, № 3, 413–444 (2012). DOI: https://doi.org/10.1216/JCA-2012-4-3-413

J. Rosales, Commutative monoids with zero-divisors, Boll. Unione Mat. Ital., 5, № 3, 773–788 (2002).

J. M. Howie, Fundamentals of semigroup theory, Oxford University Press, No.~12 (1995). DOI: https://doi.org/10.1093/oso/9780198511946.001.0001

D. H. Adams, Semigroups with no nonzero nilpotent elements, Math. Z., 123, № 2, 168–176 (1971). DOI: https://doi.org/10.1007/BF01110115

R. Giri, A. Wazalwar, Prime ideals and prime radicals in non-commutative semigroups, Kyungpook Math. J., 33, № 1, 37–48 (1993).