MDS codes have the highest possible error-detecting and error-correcting capability among codes of given length and size. Let <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> be any prime, and <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> be positive integers. Here, we consider all constacyclic codes of length <inline-formula> <tex-math notation="LaTeX">$p^{s}$ </tex-math></inline-formula> over the ring <inline-formula> <tex-math notation="LaTeX">$\mathcal R= \mathbb F_{p^{m}}+ u \mathbb F_{p^{m}}\, (u^{2}=0)$ </tex-math></inline-formula>. The units of the ring <inline-formula> <tex-math notation="LaTeX">$\mathcal R$ </tex-math></inline-formula> are of the form <inline-formula> <tex-math notation="LaTeX">$\alpha +u\beta $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\alpha, \, \beta, \, \gamma \in \mathbb F_{p^{m}}^{*}$ </tex-math></inline-formula>, which provides <inline-formula> <tex-math notation="LaTeX">$p^{m}(p^{m}-1)$ </tex-math></inline-formula> constacyclic codes. We acquire that the <inline-formula> <tex-math notation="LaTeX">$(\alpha + u \beta)$ </tex-math></inline-formula>-constacyclic codes of <inline-formula> <tex-math notation="LaTeX">$p^{s}$ </tex-math></inline-formula> length over <inline-formula> <tex-math notation="LaTeX">$\mathcal R$ </tex-math></inline-formula> are the ideals <inline-formula> <tex-math notation="LaTeX">$\langle (\alpha _{0}\,\,x - 1)^{j} \rangle $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$0 \leq j \leq 2~p^{s}$ </tex-math></inline-formula>, of the finite chain ring <inline-formula> <tex-math notation="LaTeX">$R [x] / \langle x^{p^{s}} - (\alpha + u \beta) \rangle $ </tex-math></inline-formula> and the <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>-constacyclic codes of <inline-formula> <tex-math notation="LaTeX">$p^{s}$ </tex-math></inline-formula> length over <inline-formula> <tex-math notation="LaTeX">$\mathcal R$ </tex-math></inline-formula> are the ideals of the ring <inline-formula> <tex-math notation="LaTeX">$\mathcal R[x] / \langle x^{p^{s}} - \gamma \rangle $ </tex-math></inline-formula> which is a local ring with the maximal ideal <inline-formula> <tex-math notation="LaTeX">$\langle u, x - \gamma _{0} \rangle $ </tex-math></inline-formula>, but it is not a chain ring. In this paper, we obtain all MDS symbol-pair constacyclic codes of length <inline-formula> <tex-math notation="LaTeX">$p^{s}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathcal R$ </tex-math></inline-formula>. We deduce that the MDS symbol-pair constacyclic codes are the trivial ideal <inline-formula> <tex-math notation="LaTeX">$\langle 1 \rangle $ </tex-math></inline-formula> and the Type 3 ideal of <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>-constacyclic codes for some particular values of <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>. We also present several parameters including the exact symbol-pair distances of MDS constacyclic symbol-pair codes for different values of <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>.
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