this Section are taken from [@CDG]. Let $A$ be a ring without unit. $e\in A$ is called a preunit if $ea=ae=ae^2$, for all $a\in A$. Then map $p:\ A\to A$, $p(a)=ae$, satisfies the following properties: $p\circ p=p$ and $p(ab)=p(a)p(b)$. Then $\ol{A}={\rm Coim}(p)$ is a ring with unit $\ol{e}$ and $\ul{A}={{\rm Im}\,}(p)$ is a ring with unit $e^2$. $p$ induces a ring isomorphism $\ol{A}\to \ul{A}$.\
Let $k$ be a commutative ring, $A$, $B$ $k$-algebras with unit, and $R:\ B\ot A\to A\ot B$ a $k$-linear map. We use the notation $${\label{eq:smash}}
R(b\ot a)=a_R\ot b_R=a_r\ot b_r,$$ where the summation is implicitely understood. $A\#_RB$ is the $k$-algebra $A\ot B$ with newly defined multiplication $$(a\# b)(c\# d)= ac_R\# b_Rd.$$ $(A,B,R)$ is called a weak smash product structure if $A\#_RB$ is an associative $k$-algebra with preunit $1_A\# 1_B$. The multiplication is associative if and only if $$R(bd\ot a)=a_{Rr}\ot b_rd_R~~{\rm and}~~
R(b\ot ac)=a_Rc_r\ot b_{Rr},$$ for all $a,c\in A$ and $b,d\in B$. $1_A\# 1_B$ is a preunit if and only if $$R(1_B\ot a)=a(1_A)_R\ot (1_B)_R~~{\rm and}~~R(b\ot 1_A)=(1_A)_R\ot (1_B)_Rb.$$

A left-right weak entwining structure is a triple $(A,C,\psi)$, where $A$ is an algebra, $C$ is a coalgebra, and $\psi:\ A\ot C\to A\ot C$ is a $k$-linear map satisfying the conditions 