I am an assistant professor of Mathematics and Statistics department, Science college at the University of Imam Mohammad Ibn Saud in Saudi Arabia since 2023.

I started my PhD journey in 2018, under the supervision of Prof. Vladimir Bavula. Our research is devoted to studying classifications of Poisson prime ideals for a certain class of Poisson polynomial algebras in three variables. The study of such algebras was first introduced by Oh in 2006, $[$Oh, 2006$]$. The algebras ${A}=K[t][x,y]$ are the Poisson polynomial algebras in one variable $t$, with trivial Poisson bracket, over an algebraically closed field $K$ of characteristic zero, that are extended into two variables $x$ and $y$, under certain conditions, such that if $u$ is a fixed polynomial in $K[t]$, $f$ is an arbitrary polynomial in $K[t]$, $\lambda $ is a unit element in $K^\times,$ $c$ is an arbitrary element in $ K,$ the partial derivations $\partial_t=\frac{d}{dt}$ and $\partial_y=\frac{d}{dy}$. The Poisson algebras ${A}$ can be denoted either by

$$ (K[t]; f \partial_{t}, {\lambda^{-1}}f\partial_{t}, c, u) \ \text{or}\ \ K[t][y; f \partial_{t}] [x; {\lambda^{-1}}f\partial_{t}, u\partial_{y}] $$

with Poisson bracket defined by the rule

$$ \lbrace t, y \rbrace=fy, \ \ \lbrace t, x \rbrace={\lambda^{-1}} f x \ \ \text{and}\ \ \lbrace y, x \rbrace=cyx+u.$$ The class of Poisson algebras ${A}$ splits into three classes: I, II and III. Each of them splits further into subclasses. We are interested in the Poisson spectra, minimal Poisson ideals and maximal Poisson ideals of Poisson algebras ${A}$ in order to study some properties of their Poisson algebras, Poisson enveloping algebras and some simple finite dimensional Poisson $A$-modules. The results, i.e. the containment of Poisson prime ideals for Poisson algebras that belong to some subclasses are presented in diagrams.