The Minus Partial Order on Endomorphism Rings

Year 2023, , 110 - 117, 30.06.2023

Tufan Özdin

https://doi.org/10.47000/tjmcs.1214202

Abstract

Let $S=End(M)$ be the ring of endomorphisms of a right $R$-module M. In this paper we define the minus parital order for the endomorphism ring of modules. Also, we extend study of minus partial order to the endomorphism ring of a (Rickart) module. Thus several well-known results concerning minus partial order are generalized.

Keywords

Endomorphism ring, minus partial order operator, Rickart ring

References

• Anderson, F.W., Fuller, K.R., Rings and Categories of Modules, New York: Springer-Verlag, 1974.
• Anderson Jr., W.N., Shorted operators, SIAM Journal on Applied Mathematics, 20(3)(1971), 522–525.
• Anderson, W.N., Trapp, G.E., Shorted operators II, SIAM Journal on Applied Mathematics, 28(1)(1975), 60—71.
• Djordjevic, D.S., Rakic, D.S., Marovt, J., Minus partial order in Rickart rings, Publ. Math. Debrecen, 87(3-4)(2015), 291–305.
• Hartwig, R.E., How to partially order regular elements, Math. Japonica,25(1980), 1–13.
• Jain, S.K., Blackwood, B., Prasad, K.M., Srivastava, A. K., Shorted operators relative to a partial order in a regular ring, Comm. Algebra, 37(11)(2009), 4141–4152.
• Jain, S.K., Prasad, K.M., Right-left symmetry of aR  bR = (a + b)R in regular rings, J. Pure Appl. Algebra,133(1-2)(1998), 141-–142.
• Kasch, F., Mader, A., Regularity and Substructures of Hom, Frontiers in Mathematics Basel, Switzerland: Birkh¨auser, 2009.
• Lee, G., Rizvi, S.T., Roman, C.S., Rickart modules, Comm. Algebra, 38(11)(2010), 4005–4027.
• Marovt, J., On partial orders in Rickart rings, Linear and Multilinear Algebra, 63(9)(2015), 1707–1723.
• Mitra, S.K., The minus partial order and shorted matrix, Linear Algebra Appl., 83(1986), 1–27.
• Mitsch, H., A natural partial order for semigroups, Proc. Am. Math. Soc., 97(3)(1986), 384—388.
• Nambooripad, K. S. S., The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc., 23(3)(1980), 249–260.
• Quynh, T.C., Abyzov, A., Koşan, M.T., On (unit-)regular morphisms, Lobachevskii Journal of Mathematics, 40(12)(2019), 2103–2110.
• Quynh, T.C., Koşan, M.T., Hai, P.T., A note on regular morphisms, Ann. Univ. Sci. Budapest. Sect. Comput., 41(2013), 249–260.
• Quynh, T.C., Koşan, M.T., Thuyet, L.V., On (semi)regular morphisms, Comm. Algebra, 41(8)(2013).
• Semrl, P., Automorphisms of B(H) with respect to minus partial order, J. Math. Anal. Appl., 369(1)(2010), 205–213.
• Ungor, B., Halicioglu, S., Harmanci, A., Marovt, J., Minus partial order in regular modules, Comm. Algebra, 48(10)(2020), 4542–4553.
• Ungor, B., Halicioglu, S., Harmanci, A., Marovt, J., On properties of the minus partial order in regular modules, Publ. Math. Debrecen, 96(1-2)(2020), 149–159.
• von Neumann, J., On Regular ring, Proc. Nat. Acad. Sci., 22(12)(1936), 707–713.