ESTONIAN ACADEMY
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eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
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proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
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Morita contexts and unitary ideals of rings; pp. 122–134

Full article in PDF format | 10.3176/proc.2021.2.02

Authors
Kristo Väljako, Valdis Laan

Abstract

In this paper we study Morita contexts between rings without identity. We prove that if two associative rings are connected by a Morita context with surjective mappings, then these rings have isomorphic quantales of unitary ideals. We also show that the quotient rings by ideals that correspond to each other under that isomorphism are connected by a Morita context with surjective mappings. In addition, we consider how annihilators and two-sided socles behave under that isomorphism. 


References

1. Abrams, G. D. Morita equivalence for rings with local units. Commun. Algebra, 1983, 11(8), 801–837.
https://doi.org/10.1080/00927878308822881

2. Anh, P. N. and Márki, L. Morita equivalence for rings without identity. Tsukuba J. Math., 1987, 11(1), 1–16.
https://doi.org/10.21099/tkbjm/1496160500

3. Buys, A. and Kyuno, A. Two-sided socles of Morita context rings. Chin. J. Math., 1993, 21(2), 99–108.

4. Chen, Y. Q., Fan, Y. and Hao, Z. F. Ideals in Morita rings and Morita semigroups. Acta Math. Sin., 2005, 21, 893–898.
https://doi.org/10.1007/s10114-004-0427-y

5. El Kaoutit, L. Wide Morita contexts in bicategories. Arab J Sci. Eng., 2008, 33(2C), 153–173.

6. García, J. L. and Marín, L. Morita theory for associative rings. Commun. Algebra, 2001, 29(12), 5835–5856.
https://doi.org/10.1081/AGB-100107961

7. García, J. L. and Simón, J. J. Morita equivalence for idempotent rings. J. Pure Appl. Algebra, 1991, 76(1), 39–56.
https://doi.org/10.1016/0022-4049(91)90096-k

8. Grandis, M. Homological Algebra in Strongly Non-Abelian Settings. World Scientific Publishing Co. Pte. Ltd., Singapore, 2013.
https://doi.org/10.1142/8608

9. Kashu, A. I. The composition of dualities in a nondegenerate Morita context. J. Pure Appl. Algebra, 1998, 133(1–2), 143–149. 
https://doi.org/10.1016/s0022-4049(97)00190-4

10. Laan, V., Márki, L. and Reimaa, Ü. Lattices and quantales of ideals of semigroups and their preservation under Morita contexts. Algebra Universalis, 2020, 81(24). 
https://doi.org/10.1007/s00012-020-0650-0

11. Laan, V. and Väljako, K. Enlargements of rings. Commun. Algebra, 2020, 49(4), 1764–1772.
https://doi.org/10.1080/00927872.2020.1851702

12. Lam, T. Y. Matrix rings, categories of modules, and Morita theory. In Lectures on Modules and RingsGraduate Texts in Mathematics, Vol. 189. Springer, New York, NY, 1999.
https://doi.org/10.1007/978-1-4612-0525-8_7

13. Rosenthal, K. I. Quantales and Their Applications. Longman Group UK Ltd., London, 1990. 

14. Steinfeld, O. Quasi-Ideals in Rings and Semigroups. Akadémiai Kiadó, Budapest, 1978.

15. Stenström, B. Radicals and socles of lattices. Arch. Math., 1969, 20, 258–261.
https://doi.org/10.1007/BF01899296

16. Tominaga, H. On s-unital rings. Math. J. Okayama Univ., 1976, 18, 117–134.


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