Let S = {0, 2, 4, 6, 8} `sub` Z10 with addition and multiplication as defined in Z10. a.) Construct addition and multiplication tables for S, using the operations as defined in Z10. b.)    Is S a subring of Z10 ? If not give a reason. c.)    Does S have zero divisors? d.)   Which elements of S have multiplicative inverse?

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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` .

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