We performed muon spin rotation/relaxation measurements to investigate the magnetic behavior of $T^{prime}$-type Eu$_2$CuO$_4$ (ECO), which is the parent compound of electron-doped cuprate superconductors, and the effects of oxygen-reduction annealing on its magnetism. In as-sintered (AS) ECO, we clarified the development of magnetic correlations upon cooling below $T_mathrm{N1}$ (= 265 K) as well as the coexistence of a dominant fluctuating spin state and partially ordered spin state in the temperature range between $sim$150 K and $T_mathrm{N1}$. Upon further cooling, uniform long-range magnetic order was observed below $T_mathrm{N2} = 110$ K, which is close to the ordering temperature of 115 K in $T^{prime}$-type La$_2$CuO$_4$ (LCO) [Phys. Rev. B {bf 82}, 180508(R) (2010)]. For oxygen-reduction-annealed ECO, a similar ordering sequence with the same $T_mathrm{N2}$ was observed but without the partially ordered spin state. Therefore, the fluctuating spin state over a wide temperature range and a $T_mathrm{N2}$ less than the N{e}el temperature ($T_mathrm{N2} approx T_mathrm{N1}$) in $T$-type LCO are common features of the $T^{prime}$-type parent $R_2$CuO$_4$ ($R$CO, $R$: rare-earth ion). The origin of the partially ordered spin state in AS ECO is discussed from the viewpoint of chemical defect. Furthermore, we discuss the roles of electron doping and repairing defect in the observed effect of annealing on the magnetism of $T^{prime}$-type $R$CO.
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