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

$$\{t,y\}=fy,\{t,x\}={\lambda }^{-1}fx\text{and}\{y,x\}=cyx+u.$$ 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.