Ring theory(addition table)

Question

(S,+,.) is a ring , where S={a,b,c,d}. Complete the table. $$ \begin{array}{c|ccccc} + & a & b & c & d \\ \hline a &a &b &c &d \\ b &b &1 &2 &3\\ c &c &4 &5 &a\\ d &d &6 &7 &8 \end{array} $$
The missing cells can be numbered as 1,2,3,4,5,6,7 from left to right. $$ 1=2b\\ 2=b+c\\ 3=b+d\\ 4=b+c\\ 5=2c\\ 6=d+b\\ 7=d+c\\ 8=2d\\ $$ From the table the identity is a and since $c+d=a$ ,$d=c$ inverse and $c= d$ inverse.

I think all the answers should be a single letter but I can't simplify any further with the information given. Any type of help would be appreciated.

Answer 1

Hint:

In a ring, the addition table is the table of a group. What are the groups of order $4$?

Partial solution:

There are two groups of order $4$: $C_4$ and $C_2 \times C_2$

Full solution:

There are two rings of size $4$: $\mathbb Z_4$ and $\mathbb Z_2 \times \mathbb Z_2$