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Let $\mathbb{F}_{p^{m}}$ be a finite field with cardinality $p^{m}$ and$R=\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ with $u^{2}=0$. We aim to determineall $\alpha+u\beta$-constacyclic codes of length $np^{s}$ over $R$, where$\alpha,\beta\in\mathbb{F}_{p^{m}}^{*}$, $n, s\in\mathbb{N}_{+}$ and$\gcd(n,p)=1$. Let $\alpha_{0}\in\mathbb{F}_{p^{m}}^{*}$ and$\alpha_{0}^{p^{s}}=\alpha$. The residue ring $R[x]/\langlex^{np^{s}}-\alpha-u\beta\rangle$ is a chain ring with the maximal ideal$\langle x^{n}-\alpha_{0}\rangle$ in the case that $x^{n}-\alpha_{0}$ isirreducible in $\mathbb{F}_{p^{m}}[x]$. If $x^{n}-\alpha_{0}$ is reducible in$\mathbb{F}_{p^{m}}[x]$, we give the explicit expressions of the ideals of$R[x]/\langle x^{np^{s}}-\alpha-u\beta\rangle$. Besides, the number ofcodewords and the dual code of every $\alpha+u\beta$-constacyclic code areprovided.

All $\alpha+u\beta$-constacyclic codes of length $np^{s}$ over $\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$