So I have a solution I want to see if my thought process is correct...
The ring is a commutative ring with identity but is not an integral domain. The units of the ring are $\pm1= 1,\;3,\;7,\;9$. I got these values from a multiplication table of elements of $\mathbb Z/10\mathbb Z$
Next we see $2=2\cdot6, 4=2\cdot7, 5=3\cdot5, 6=2\cdot8, 8=2\cdot9$. Since $3$ and $9$ are units than $5$ and $9$ are irreducibles.
Finally we know $2,\;4,\;6,\;8$ divides $0,\;2,\;4,\;6,\;8$ and $5$ divides $0,\;5$ so we see for example if $2\mid ab$ then $2\mid a$ or $2\mid b$ so all these elements are all primes.
If this is not the correct way of thinking is there is a better way let me know!