a)

addition table in `S`

0+0 =0 2+0 =2 4+0 =4 6+0 =6 8+0 =8

0+2 =2 2+2 =4 4+2 =6 6+2 =8 8+2 =0

0+4 =4 2+4 =6 4+4 =8 6+4 =0 8+4 =2

0+6 =6 2+6 =8 4+6 =0 6+6 =2 8+6 =4

0+8 =8 2+8 =0 4+8 =2 6+8 =4 8+8 =6

multiplication table in `S`

0*0 =0 2*0 =0 4*0 =0 6*0 =0 8*0 =0

0*2 =0 2*2 =4 4*2 =8 6*2 =2 8*2 =6

0*4 =0 2*4 =8 4*4 =6 6*4 =4 8*4 =2

0*6 =0 2*6 =2 4*6 =4 6*6 =6 8*6 =8

0*8 =0 2*8 =6 4*8 =2 6*8 =8 8*8 =4

b)

Since S =2*`NN5` and ` ``NN5` is a ring itself, then the answer to this question is yes, S is a subring of `ZZ10`

c) From the multiplication table one can see that there is no nonzero element a in `S` which multiplied with a nonzero element b in gives `a*b =0` . Thus there are no zero divisors in `S`

d) `S` does not contain 1 element. Thus the 1 element does not appear in the multiplication table as a result of any `a*b` , where `a,b in S`. Therefore there are no elements that have multiplicative inverses in `S` .

**Further Reading**