The flavor changing neutral current decays $$t\rightarrow c X$$ $t\to cX$ ($$X=\gamma ,\,g,\, Z,\, H$$ $X=\gamma ,g,Z,H$ ) and $$t\rightarrow c{{\bar{\ell }}}\ell $$ $t\to c\overline{\ell }\ell$ ($$\ell =\mu ,\,\tau $$ $\ell =\mu ,\tau$ ) are studied in a renormalizable scalar leptoquark (LQ) model with no proton decay, where a scalar SU(2) doublet with hypercharge $$Y=7/6$$ $Y=7/6$ is added to the standard model, yielding a non-chiral LQ $$\varOmega _{5/3}$$ ${\Omega }_{5/3}$ . Analytical results for the one-loop (tree-level) contributions of a scalar LQ to the $$f_i\rightarrow f_j X$$ $f_i\to f_{j}X$ ($$f_i\rightarrow f_j {\bar{f}}_m f_l$$ $f_i\to f_j{\overline{f}}_m{f}_l$ ) decays, with $$f_a=q_a, \ell _a$$ $f_a=q_a,{\ell }_a$ , are presented. We consider the scenario where $$\varOmega _{5/3}$$ ${\Omega }_{5/3}$ couples to the fermions of the second and third families, with its right- and left-handed couplings obeying $$\lambda _R^{\ell u_i}/\lambda _L^{\ell u_i}=O(\epsilon )$$ ${\lambda }_R^{\ell u_i}/{\lambda }_L^{\ell u_i}=O\left(ϵ\right)$ , where $$\epsilon $$ $ϵ$ parametrizes the relative size between these couplings. The allowed parameter space is then found via the current constraints on the muon $$(g-2)$$ $(g-2)$ , the $$\tau \rightarrow \mu \gamma $$ $\tau \to \mu \gamma$ decay, the LHC Higgs boson data, and the direct LQ searches at the LHC. For $$m_{\varOmega _{5/3}}=1$$ $m_{{\Omega }_{5/3}}=1$ TeV and $$\epsilon =10^{-3}$$ $ϵ={10}^{-3}$ , we find that the $$t\rightarrow c X$$ $t\to cX$ branching ratios are of similar size and can be as large as $$10^{-8}$$ ${10}^{-8}$ in a tiny area of the parameter space, whereas $${\mathrm{Br}}(t\rightarrow c{{\bar{\tau }}}\tau )$$ $\mathrm{Br}(t\to c\overline{\tau }\tau )$ [$${\mathrm{Br}}(t\rightarrow c{{\bar{\mu }}}\mu )$$ $\mathrm{Br}(t\to c\overline{\mu }\mu )$ ] can be up to $$10^{-6}$$ ${10}^{-6}$ ($$10^{-7}$$ ${10}^{-7}$ ).