Decidability of logical theories

ᴬ Is the statement $\exists x \forall y[x+y=y]$ a member of $\operatorname {Th}(\mathcal{N},+)$? Why or why not? What about the statement $\exists x \forall y[x+y=x]$?

ᴬ Let $(\mathcal{N},<)$ be the model with universe $\mathcal{N}$ and the "less than" relation. Show that $\operatorname {Th}(\mathcal{N},<)$ is decidable.

For each $m>1$ let $\mathcal{Z}_{m}=\left\{ 0,1,2, \ldots , m-1\right\}$, and let $\mathcal{F}_{m}=\left(\mathcal{Z}_{m},+, \times \right)$ be the model whose universe is $\mathcal{Z}_{m}$ and that has relations corresponding to the + and × relations computed modulo $m$. Show that for each $m$, the theory $\operatorname {Th}\left(\mathcal{F}_{m}\right)$ is decidable.

Let $R \subseteq \mathcal{N}^{k}$ be a $k$-ary relation. Say that $R$ is definable in $\operatorname {Th}(\mathcal{N},+)$ if we can give a formula $\phi$ with $k$ free variables $x_{1}, \ldots , x_{k}$ such that for all $a_{1}, \ldots , a_{k} \in \mathcal{N}$, $\phi \left(a_{1}, \ldots , a_{k}\right)$ is true exactly when $a_{1}, \ldots , a_{k} \in R$. Show that each of the following relations is definable in $\operatorname {Th}(\mathcal{N},+)$.