Let $(A,+, \cdot)$ be a ring with the given tables for operations $+$ and $\cdot$
\begin{array}{c|cccc} + & s & t & x & y \\ \hline s & y & x & s & t \\ t & x & y & t & s \\ x & s & t & x & y\\ y & t & s & y & x\end{array}
\begin{array}{c|cccc} \cdot & s & t & x & y \\ \hline s & y & y & x & x \\ t & y & y & x & x \\ x & x & x & x & x\\ y & x & x & x & x\end{array}
determine: a) which is the zero of the ring b) the inverse of each element;
for a) what I can see is that $x$ act as element $0$ for $+$
also $x$ act as element $0$ for $\cdot$ but for $y$ the result equals $x$ too
for b) $s+t=x$ so if $x=0$ then $s$ is inverse of $t$
in $\cdot$ I can't see how is $s\cdot s= s \cdot t =y$, because as said before $s$ and $t$ is inverse of each other...
I would appreciate if someone can give some insight on how to solve this problem
$x$ is the additive identity.
The additive inverses are $-s=t,-t=s,-x=x,-y=y$.
This isn't a ring with unit.