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The classes of 1MP-inverses and MP1-inverses are recently introduced classes of generalized inverses of complex matrix. Actually, they coincide with the classes of $\{1,2,3\}$ and $\{1,2,4\}$ inverses, respectively. We consider these inverses in the context of a ring with involution and prove that their most important characterizations and properties remain true. We show that the binary relations based on these inverses are in fact the well known left-star and right-star partial orders. We extend these relations to the ring case, connect them with the unified theory of partial order relations based on generalized inverses and provide several properties. Finally, we indicate how these results can be applied to bounded Hilbert space operators., Comment: 19 pages This manuscript was submitted in "Revista de la Real Academia de Ciencias Exactas, F\'isicas y Naturales. Serie A. Matem\'aticas" on March 21, 2022

We note that the well-known result of Von Neumann \cite{von} is not valid for all doubly substochastic operators on discrete Lebesgue spaces $\ell^p(I)$, $p\in[1,\infty)$. This fact lead us to distinguish two classes of these operators. Precisely, the class of increasable doubly substochastic operators on $\ell^p(I)$ is isolated with the property that an analogue of the Von Neumann result on operators in this class is true. The submajorization relation $\prec_s$ on the positive cone $\ell^p(I)^+$, when $p\in[1,\infty)$, is introduced by increasable substochastic operator and it is provided that submajorization may be considered as a partial order. Two different shapes of linear preservers of submajorization $\prec_s$ on $\ell^1(I)^+$ and on $\ell^p(I)^+$, when $I$ is an infinite set, are presented.

Let $R$ be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary $*$-ring case. It is shown that the group, Moore-Penrose, core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized. A number of characterizations of EP elements in $R$ are obtained. It is shown that core and dual core inverse belong to the class of inverses along an element and to the class of $(b,c)$-inverses. The case when $R$ is algebra of all bounded linear operators on Hilbert space is specifically considered.