  Let $\mathcal{R}$ be a unital ring with involution. The notions of
1MP-inverse and MP1-inverse are extended from $M_{m,n}(\mathbb{C)}$, the set of
all $m\times n $ matrices over $\mathbb{C}$, to the set $\mathcal{R}%
^{\dagger}$ of all Moore-Penrose invertible elements in $\mathcal{R}$. We study
partial orders on $\mathcal{R}^{\dagger}$ that are induced by 1MP-inverses and
MP1-inverses. We also extend to the setting of Rickart $\ast $-rings the
concept of another partial order, called the plus order, which has been
recently introduced on the set of all bounded linear operators between Hilbert
spaces. Properties of these relations are investigated and some known results
are thus generalized.
